UC-NRLF 


735 


IN  MEMORIAM 
FLORIAN  CAJOR1 


THE 

JUVENILE  ARITHMETIC!* 

AND 

SCHOLAR'S  GUIDE; 

WHEREIN  THEORY  AND  PRACTICE  ARE  COMBINED  AND 
ADAPTED  TO  THE  CAPACITIES 

OF 

YOUNG  BEGINNERS; 

CONTAINING  A  DUE  PROPORTION  OF  EXAMPLES 
IN 

FEDERAL  MONEY; 

AND  THE  WHOLE  BEING  ILLUSTRATED  BY  NUMEROUS 
QUESTIONS  SIMILAR  TO  THOSE 

w- 
OF 

PE&TALOZZL 


BY  MARTIN  RUTER,  A.  M. 

Cfncfwnatf: 

PUBLISHED  AND  SOLD  BY  N.  AND  G.  GUILFORD 


.  AND  O.  FARNSWORTH,  JR.  PRINTERS 

1828, 


DISTRICT  OF  OHIO,  TO  WIT : 

BE  IT  REMEMBERED,  That  on  the  twenty-second  day  of  April,  in  the 
year  of  our  Lord  one  thousand  eight  hundred  and  twenty  seven,  and  in  the  fifty- 
first  year  of  the  American  Independence,  MARTIN  RUTER,  of  said  District, 
hath  deposited  in  said  office  the  title  of  a  book,  the  right  whereof  he  claims  as  au- 
thor and  proprietor  in  the  words  following,  to  wit: 

"THE  JUVENILE  ARITHMETICS:  AND    SCHOLAR'S  GUIDE: 

wherein  theory  and  practice  are  combined  and  adapted  to  the  capacities  of  young 
beginners;  containing"  a  due  proportion  of  examples  in  Federal  Money,  and  the 
•whole  being  illustrated  by  numerous  questions  similar  to  those  of  PESTALOZ- 
ZI,by  MARTIN  RUTER,  A.  JI/;" 

In  conformity  to  the  Act  of  the  Congress  of  the  United  States,  entitled  "  An  Act 
for  the  encouragement  of  Learning,  by  securing  the  copies  of  Maps,  Charts,  and 
Books,  to  the  Authors  and  Proprietors  of  such  Copies,  during  the  times  therein 
mentioned;  and,  also,  of  the  Act  entitled  "  AH  Act  supplementary  to  an  Act  enti- 
tled an  Act  for  the  encouragement  of  Learning,  by  securing  the  copies  of  Man>, 
Charts,  and  Books  to  the  Authors  and  Proprietors  of  such  Copies  during  the  times 
therein  mentioned,  and  extending  the  benefits  thereof  to  the  Arts  of  designing,  en- 
graving, and  etching  historical  and  other  Prints. 

WM.  KEY  BOND, 
Clerk  of  the  District  of  Ohio. 


RECOMMENDATIONS. 

The  following  have  been  selected  from  the  recommendations  be* 
stowed  upon  this  work. 

Messrs.  Guilfords, — I  have  examined  hastily  the  u  Juvenile  Arithmetick,"  which 
you  sent  me,  and  am  of  opinion  that  it  possesses  some  advantages  over  those  gen- 
erally in  use:  I  particularly  refer  to  the  part  intended  to  Cultivate  in  the  learner, 
the  habit  of  going  through  the  solutions  mentally.  Very  respectfully  yours, 

JOHN  E.  ANNAN, 

Professor  of  Mathematicks  and  Natural  Philosophy  in  the  Miami  University. 
Oxford,  June  5, 1827. 


From  a  hasty  review  of  Dr.  Ruter's  Arithmetick,  I  am  inclined  to  think  well  of 
it.     The  attempt  to  introduce  a  rational  method  of  instruction  in  arty  department 
of  education,  is  laudable  and  especially  in  common  schools.    This  I  think  the  Ju-  Ilf 
venile  Arithmetick  is  well  calculated  to  do,  in  that  branch  of  study  to  which  it  be-  " 
longs.    The  plan  of  Pestalozzi  is  excellent,  and  Dr.  Ruter  has  perhaps  imitated  it 
more  successfully  (by  comprizing  more  in  less  space)  thanMr.  Colburnj  between,. 
whose  Aritfrnietick  and  this  there  is  however  a  c 


Tour's,  &c. 

WM.  H.  M'GUFFY, 

Professor  of  Languages,  &c.,  in  the  Miami  University. 
Orford,  June  28, 1827. 


We  havo  examined  your  'Juvenile  Arithmetick,'  and  feel  a  pleasure  in  recom- 
nending  it  to  the  schools  of  our  country.  We  think  the  general  arrangement  good, 
ind  have  no  hesitation  in  saying,  that  the  questions  prefixed  and  appended  to  the 
•ules,  give  it  superior  advantages.  Respectfully  yours, 

JOSEPH  S.  TOMLINSON, 
JOHN  P.  DURBIN, 
March  12,  1823-  Professors  in  Augusta  College,- 


The  Juvenile  Arithmetick,  from  the  cursory  examination  which  I  have  given  it, 
appears  to  be  a  manuel  of  value  for  the  introduction  of  youth,  into  the  science  of 
numbers.  In  furnishing  a  second  edition,  I  wish  you  success. 

ELIJAH  SLACK. 

I  concur  most  cheerfully  in  th«  above  opinion.  S.  JOHNSTON. 

Cincinnati,  April  2, 1828. 


I  have  used  thy  compilation  of  Arithmetick  during  the  last  year;  and  do  not  hefcP- 
fate  in  recommending  it  to  the  publick.  The  questions  pieceding  the  rules,  the 
particular  attention  to  fractions,  and  the  sketch  of  mensuration  give  it  a  decidetl 
preference  to  any  other  here  in  use.  JOHN  L.  T ALBERT. 

Cincinnati,  Fourth  mo,  5,  1828. 

Having  examined  the  above  Arithmetick,  I  cheerfully  concur  in  the  foregoing 
•pinion  of  its  merits.  ARNOLD  TKUESDBLI.. 


M306089 


IV  RECOMMENDATIONS. 


I  have  carefally  inspected  "the  Juvenile  Arithmetick  and  Scholars  Guide,"  by 
Dr.  Ruter  and  am  of  the  opinion,  it  is  well  calculated  and  arranged,  to  conduct  thk 
pupil  by  an  easy  gradation  to  a  perspicuous  conception  of  the  science  of  numbers- 
I  therefore  recommend  it  to  publick  nse,  particularly  in  common  schools. 

SAMUEL  BURR, 

September  2, 1827.  .  Professor  of  Mathernaticks 


A  cursory  examination  of  Dr.  M.  Ruler's  Arithmetick,  has  convinced  me,  that 
the  simple  and  familiar  manner  in  which  the  learned  author  unfolds  the  principles 
of  this  science,  and  adapts  them  ttr  the  understanding  of  the  young  learner,  can 
not  fail  to  give  his  work  a  decided  preference,  for  practical  purposes,  over  those 
arhhmeticks  in  common  use.  In  my  opinion,  teachers  who  adopt  it,  as  well  as 
pupils  who  study  it,  wilLrealize  satisfactory  and  highly  beneficial  results. 

S.  KIRKHAM, 

Pittsburgh^  April  2, 1823.  Author  of  Grammar  in  Familiar  Lectures. 


I  have  examined  the  system  of  Arithmetick  compiled  by  Dr.  Ruter,  and  am  of 
opinion  that  it  is  well  calculated  for  conveying  to  youth,  a  general  knowledge  of 
that  science  in  a  shorter  time,  than  any  I  have  seen. 

G.  GARDNER, 

March  29, 1828.  Teacher  of  Mathematicks,  Mill-Creek  Township-. 


Having  examined  the  Juvenile  Arithmetick,  I  have  no  hesitation  in  pronouncing 
it  an  excellent  elementary  School  Book.  The  rules  are  Judiciously  arranged,  and 
peculiarly  well  adapted  to  juvenile  comprehension:  The  work  contains  multnm  in 
parvo,  and  I  think  its  publication  will  be  conducive  to  publick  utility. 

Hoping  its  merits  will  be  duly  appreciated,  I  take  great  pleasure  in  recommen- 
ding it  to  the  publick  patronage.  Your's  respectfully, 

RICHARD  MORECRAFT, 

Cincinnati,  January  2,  1828.  Teacher- 


From  my  acquaintance  with  Ruler's  Arithmetick,  I  am  convinced  that  it  is  well 
calculated  to  encourage  the  student,  improve  his  mind,  and  prepare  him  for  busi- 
ness. JOHN  LOCKE, 

May  16, 1828.  Principal  of  Cincinnati  Female  Academy. 


Gentlemen — I  have  examined  with  some  attention  the  Juvenile  Arithmetick,  &C. 
by  the  Rev.  Dr.  Ruter,  and  am  decidedly  of  opinion,  that  it  is  admirably  calcula 
ted  for  conveying  to  youth  with  great  facility  a  general  knowledge  of  that  impor 
rant  science.  Tlie  ingenious  manner  in  which  the  compiler  has  given  an  elucida- 
tion of  Vulgar  Fractions,  together  with  an  exclusion  of  all  extraneous  matter, 
fenders  it  in  my  estimation  a  treatise  of  peculiar  merit. 
Your  obedient  servant, 

JOHN  WINRIGHT, 
Cincinnati,  September  2, 1827 .  Teacher 


PREFACE. 


THIS  ARITHMETICK  has  been  compiled  with  a  view 
to  facilitate  the  progress  of  pupils,  and  lessen  the  la- 
bour of  teachers.  The  questions  preceding  and  fol- 
lowing the  rules,  are  designed  to  lead  young  learners 
into  habits  of  thinking  and  calculating;  and  thus, 
to  prepare  them  for  practical  operations.  Experi- 
ence has  demonstrated,  that,  in  the  instruction  of 
children  in  any  science,  it  is  necessary  to  excite  their 
entire  attention  to  the  subject  before  them.  The  la- 
tent energies  of  their  minds  must  be  roused  up,  and 
called  forth  into  action.  When  this  can  be  effectually 
done,  success  is  rendered  certain.  To  accomplish 
this  important  object,  the  best  method  has  been  found 
in  the  frequent  use  of  well  selected  questions.— 
Though  it  is  a  successful  course  in  all  juvenile  stu- 
dies, it  is  particularly  so  in  the  science  of  numbers; 
and  the  progress  of  pupils  must  be  slow  without  it. 
The  questions  in  the  following  pages  are  thought  to 
be  sufficiently  numerous  for  the  purposes  intended; 
the  rules  have  been  arranged  according  to  the  plan 
of  some  of  the  best  authors  on  this  subject,  and  the 
work  is  offered  to  the  publick  with  the  hope  that  it 
will  be  useful  in  the  schools  of  our  country. 

M.  R. 


EXPLANATION   OF   THE   CHARACTERS  USED  IN 

ARITHMETICS:. 


4-         Signifies  plus,  or  addition. 

Signifies  minus,  or  subtraction. 
X         Denotes  multiplication. 
-T-         Means  division. 
:   :  '  :  Signifies  proportion. 
=         Denotes  equality. 

Thus,  4+7  denotes  that  7  is  to  be  added  to  4. 
5 — 3,  denotes  that  3  is  to  be  taken  from  5. 
8X2,  Signifies  that  8  is  to  be  multiplied  by  2. 
9-r3,  That  9  is  to  be  divided  by  3. 
3  :  2  :  :  6  :  4,  Shows  that  3  is  to  2  as  6  is  to  4. 
7-j-9=  16,  Shows  that  the  sum  of  7  and  9  is  equal  to  16, 
^/  or  I/  Denotes  the  Square  Root, 
s/  Denotes  the  Cube  Root. 
{/  Denotes  the  Biquadrate  Root. 

This  mark,  called  a  Vinculum,  shows  that  the 

several  figures  over  which  it  is  drawn  are  to 
he  fcken  together  as  a  simple  quantity. 


ARITHMETICK. 


ARITHMETICK  is  the  science  which  treats  of  the  nature 
and  properties  of  numbers:  and  its  operations  are  con- 
ducted chiefly  by  five  principal  rules.  These  are,  Nu- 
meration, Addition,  Subtraction*  Multiplication,  and  Di- 
vision. 

Numbers  in  Arithmetick  are  expressed  by  the  fol- 
lowing ten  digits  or  characters,  namely:  1  one,  2  two, 
3  three,  4  four,  5  five,  6  six,  7  seven,  8  eight,  9  nine, 

0  cypher. 

An  Integer  signifies  a  whole  number,  or  certain  quan- 
tity of  units,  as  one,  three,  ten.  A  Fraction  is  a  broken 
number,  or  part  of  a  number,  as  ^  one  half,  f  two-thirds, 

1  one-fourth,  f  three-fourths,  f  five-sevenths,  &c. 


NUMERATION. 

Numeration  teaches  the  different  value  of  figures  by 
their  different  places,  and  to  express  any  proposed  num- 
bers either  by  words  or  characters;  or  to  read  and 
write  any  sum  or  number. 

NUMERATION   TABLE. 

Units* 
Tens. 
Hundreds. 
Thousands, 
Tens  of  thousands. 
e<  co  *5  «3  ^  co  w   Hundreds  of  thousands, 

Millions. 

t-<Ncou5    Tens  of  millions. 
co  *-  i>  co   Hundreds  of  millions, 
G<  co  i>   Thousands  of  millions. 
10  °P    Ten  thousands  of  millions* 
^  Hundred  th&usands  of  millions., 


8  NUMERATION. 

Here,  any  figure  in  the  place  of  units,  reckoning  from 
right  to  left,  denotes  only  its  simple  value;  but  that  in 
the  second  place  denotes  ten  times  its  simple  value;  and 
that  in  the  third  place,  one  hundred  times  its  simple 
value;  and  so  on,  the  value  of  any  figure  in  each  suc- 
cessive place,  being  always  ten  times  its  former  value. 
Thus  in  the  number  6543,  the  3  in  the  first  place  denetes 
only  three;  but  4  in  the  second  place  signifies  four  tens, 
or  40;  5  in  the  third  place,  five  hundred;  and  6  in  the 
fourth  place,  six  thousand ;  which  makes  the  whole  num- 
ber read  thus — six  thousand  five  hundred  and  forty  three. 
The  cvpher  stands  for  nothing  when  alone,  or  when  on 
the  left  hand  side  of  an  integer;  but  being  joined  OB  the 
right  hand  side  of  other  figures,  it  increases  their  value 
in  the  same  ten  fold  proportion:  thus,  50  denotes  five 
tens;  and  500  is  read  five  hundred. 

Though  the  preceding  numeration  table  contains  only 
twelve  places,  which  render  it  sufficiently  large  for 
young  students,  yet  it  may  be  extended  to  more  places 
at  pleasure. 

EXAMPLE. 

Quatrillions .       Trillions.     Billions.     Millions.      Units. 
987,654;         321,234;     567,898;     765,432;  123,456 

Here  note,  that  Billions  is  substituted  for  millions  of 
millions:  Trillions,  for  millions  of  millions  of  millions: 
Quatrillions,  for  millions  of  millions  of  millions  of  mil- 
lions. From  millions,  to  billions,  trillions,  qwatrillions, 
and  other  degrees  of  numeration,  the  same  intermediate 
denominations,  of  tens,  hundreds,  thousands,  fyc.  are  used, 
as  from  units  to  millions.  And  thus,  in  ascertaining  the 
amount  of  very  high  numbers,  we  proceed  from  Millions 
to  Billions,  Trillions,  Quatrillions,  Quintillions,  Sextil- 
lions,  Septillions,  Octillions.  Nonillions,  Decillions,  Un- 
decillions,  Duodecillions,  Tredecillions,  Quatuordecil- 
lions,  Quindecillions,  Sexdecillions,  Septendecillions, 
Octodecillions,  Novemdecillione,  VigintilHons.&c.  all  of 
which  answer  to  millions  so  often  repeated,  as  their  in- 
dices respectively  require,  according  to  the  above  pro- 
portion. 


SIMPLE   ADDITION.  0 

THE    APPLICATION. 

~Write  down,  in  figures,  the  following  numbers: 
Ten.      -  10 

Twenty-one.  -  21 

Thirty-five.    -  35 

Four  hundred  and  sixty-seven.  467 

Two  thousand  three  hundred  and  eighty-nine.  2389 
Thirty-four  thousand  five  hundred  and  seventy.  34570 
Six  hundred  and  three  thousand  four  hundred.  603400 
Seven  millions  eight  hundred  and  four  thou- )  78Q4329 

sand  three  hundred  and  t  wenty-nine.     \ 
Fifty-eight  millions  seven  hundred  and  thir-j   50739105 

ty-two  thousand  one  hundred  and  five.  \  " 
Eight  hundred  and  ten  millions  nine  hun-) 

d red  and  two  thousand  five  hundred>  810902512 
and  twelve.  ) 

Three  thousand  two  hundred  and  three) 

millions  six  hundred  and  eight  thou->  3203608999 
sand  nine  hundred  and  ninety-nine.   ) 

Question    1.   What  is  Arithmetick? 

2.  What  are  the  ten  digits  by  which  numbers 

are  expressed? 

3.  What  is  an  integer? 

4.  What  is  a  fraction? 

5.  What  are  the  principal  rules  by  which  the 

operations  in  Arithmetick  are  conducted  ? 

6.  What  does  Numeration  teach? 


SIMPLE  ADDITION. 

Simple  Addition  teaches  to  put  together  numbers  of 
the  same  demonination  into  one  sum;  as  5  dollars,  4 
dollars,  and  3  dollars,  make  12  dollars. 

Before  the  pupil  enters  upon  Addition  in  the  usual  way,  with 
figures,  it  would  be  useful  for  him  to  learn  to  perform  easy  ope- 
rations in  his  mind.  For  this  purpose  let  him  be  exercised  in 
the  following  questions,  or  in  others  which  are  similar. 


10  SIMPLE   ADDITION. 

1.  If  you  have  two  cents  in  one  hand  and  one  in  the 
other,  how  many  hare  you  in  both? 

2.  If  you  have  three  cents  in  one  hand  and  two  in 
the  other,  how  many  have  you  in  both? 

3.  If  you  have  five  cents  in  one  hand,  and  two  in  the 
other,  how  many  have  you  in  both? 

4.  John  has  six  cents,  and  Robert  has  three;  how  ma- 
ny have  they  both  together? 

5.  Charles  gave  five  cents  for  an  orange,  and  two  for 
an  apple;  how  many  did  he  give  for  both? 

6.  Dick  had  four  nuts,  John  had  three,  and  David 
had  two;  how  many  had  they  all  together? 

7.  Henry  had  five  peaches,  Joseph  had  three  and 
Tom  had  two,  and  they  put  them  all  into  a  basket;  how 
many  were  there  in  the  basket? 

8.  Three  boys,  Peter,  John  and  Oliver,  gave  some 
money  to  a  beggar.     Peter  gave  seven  cents,  John  four, 
and  Oliver  three.     How  many  did  they  all  give  him? 

9.  A  man  bought  a  sheep  for  eight  dollars,  and  a  calf 
for  seven  dollars;  what  did  he  give  for  both? 

10.  A  boy  gave  to  one  of  his  companions  eight  peaches; 
to  another  six;  to  another  four;  and  kept  two  himself; 
how  many  had  he  at  first? 

11.  How  many  are  two  and  three? — two  and  five? — 
three  and  seven? — four  and  five? 

12.  How  many  are  twice  four? — twice  five? — twice  six? 
twice  seven? — twice  eight? — twice  nine? 

13.  How  many  are  three  and  two  and  one? 

14.  How  many  are  four  and  three  and  two? 

15.  How  many  are  five  and  four  and  three? 

16.  How  many  are  four  and  five  and  two? 

17.  How  many  are  seven  and  three  and  one? 

18.  How  many  are  eight  and  four  and  two? 

19.  How  many  are  nine  and  five  and  one? 

20.  How  many  are  five  and  six  and  seven? 

21.  Ho iv  many  are  four  and  three  and  two  and  one? 

22.  How  many  are  two  and  three  and  one  and  four? 

23.  How  many  are  five  and  three  and  two  and  one?* 

*  It  is  expected  that  m;my  of  these  questions  will  be  varied  by  the  teacher,  and 
icndered  harder,  or  easier,  or  others  eiibstitutyd,  as  the  capacity  of  the  pupil  may 
require. 


SIMPLE    ADDITION.  11 

RULE. 

Place  the  figures  to  be  added,  one  under  another,  so 
that  units  will  stand  under  units,  tens  under  tens,  hun- 
dreds under  hundreds,  &c.  Draw  a  horizontal  line  un- 
der them,  and  beginning  at  the  bottom  of  the  first  column, 
on  the  right  hand  side,  that  is.  at  units,  add  together  the 
figures  in  that  column,  proceeding  from  the  bottom  to 
the  top.  Consider  how  many  tens  are  contained  in  their 
sum,  and  how  many  remain  besides  the  even  number  of 
tens,  placing  the  amount  under  the  column  of  units,  and 
carrying  so  many  as  you  have  tens  to  the  next  column. 
Proceed  in  the  same  manner  through  every  column,  set- 
ting down  under  the  last  column  its  full  amount. 

PROOF. 

Begin  at  the  top  of  the  sum  and  add  the  several  rows 
of  figures  downwards  as  they  were  added  upwards,  and 
if  the  additions  in  both  cases  be  coriect  the  sums  will 
agree. 

EXAMPLES. 
I.  II.  III.  IV. 

12  321  4000  542210 

21  123  3124  135403 

34  410  2345  350212 

10  203  5234  201304 


77      1057     14703   1229129 


V.  VI.  VII. 

2  4  0  5  6  7  £.  50678  450789 

3540210*  7  6  £'4  3  876543 

4321023*  20134  450789 

4065243*  56787  876543 

2123456  65432  234798 


12  SIMPLE   ADDITION. 

VIII.  IX.  X. 

678987G5  45  20000000 

4321234  678  3000000 

567898  9876  400000 

76543  54321  50000 

2123  234567  6000 

212  8987654  700 


72866775     9287141    23456700 


XI. 

XII. 

XIII. 

24681012 

54321231 

9  1 

i  7  6  543  2 

42130538 

19000310 

1  c< 

I  3  4  5  5  7  8 

71021346 

20304986 

9  i 

J7  6  54  3  2 

20324913 

19876540 

1  S 

>  3  4  5  6  7  8 

98765432 

98755432 

9  i 

3765432 

12345678 

12000987 

1  < 

2345678 

APPLICATION. 

1.  A  boy  owed  one  of  his  companions  6  cents;  he  owed 
another  8,  another  5,  and  another  9.     How  much  did  he 
owe  in  all?  Ans.  28  cents. 

2.  A  man  received  of  one  of  his  friends  7  dollars,  of 
another  10,  of  an  another  19,  and  of  another  50.     How 
many  dollars  did  he  receive?  Ans.  86  dollars. 

3.  A  person  bought  of  one  merchant  ten  barrels  of 
flour,  and  paid  40  dollars;  of  another  twenty  barrels  of 
cider,  for  which  he  paid  60  dollars,  and  twenty  barrels 
of  sugar  at  450  dollars;  and  of  another  ninety-five  bar- 
rels of  salt  at  570.     How  many  barrels  did  he  buy,  and 
how  much  money  did  he  pay  for  the  whole? 

Ans.  145  barrels,  and  pqd  1120  dollars. 

4.  A  had  250  dollars; 'B-had  375;*C  had  5423;  D, 
64320;  E,  287432,  and  F,  4321507.    ^ow  murh  would 
it  all  make,  if  put  together?       *    Ans'.'  4679367  dollars. 

Question   1.  What  does  Simple  Addition  teach? 

2.  How  do  you  place  the  numbers  to  be  added? 

3.  Where  do  you  begin  the  addition? 

4.  How  do  you  prove  a  sum  in  Addition? 


SIMPLE   SUBTRACTION.  13 

SIMPLE  SUBTRACTION. 

Simple  Subtraction  teaches  to  take  a  less  number  from 
*i  greater  of  the  same  denomination,  and  thus  to  find  the 
difference  between  them. 

Questions  to  prepare  the  learner  for  this  rule. 
k  If  you  have  seven  cents,  and  give  away  two;  how 
many  will  you  have  left  ? 

2.  If  you  have  eight  cents,  and  lose  four  of  them ;  how 
many  will  you  have  left? 

3.  A  boy  having  ten  cents,  gave  away  four  of  them; 
how  many  had  he  left? 

4.  A  man  owing  twelve  dollars,  paid  four  of  it;  how 
much  did  he  then  owe? 

5.  A  man  bought  a  firkin  of  butter  for  fifteen  dollars, 
and  sold  it  again  for  ten  dollars;  how  much  did  he  lose? 

6.  If  a  horse  is  worth  ten  dollars,  and  a  cew  is  worth 
four;  how  much  more  is  the  horse  worth  than  the  cow? 

7.  A  boy  had  eleven  apples  in  a  basket,  and  took  out 
five;  how  many  were  left? 

8.  Susan  had  fourteen  cherries,  and  ate  four  of  them ; 
how  many  had  she  left? 

9.  Thomas  had  twenty  cents,  and  paid  away  five  of 
them  for  some  plums;  how  many  had  he  left? 

10.  George  is  twelve  years  old,  and  William  is  seven; 
how  much  older  is  George  than  William? 

11.  Take  four  from  eight;  how  many  will  remain? 

12.  Take  three  from  nine;  how  many  will  remain? 

13.  Take  five  from  ten;  how  many  will  remain?' 

14.  Take  six  from  ten;  how  many  will  remain? 

15.  Take  six  from  eleven;  how  many  will  remain? 

16.  Take  five  from  twelve;  how  many  will  remain? 

17.  Take  four  from  thirteen;  how  many  will  remain? 

18.  Take  six  from  fourteen;  how  many  will  remain? 

19.  Take  six  from  fifteen;  how  many  will  remain? 

20.  Take  eight  from  sixteen;  how  many  will  remain? 

21.  Take  nine  from  twelve;  how  many  will  remain? 

22.  Take  nine  from  thirteen;  how  many  will  remain? 

23.  Take  three  from  thirteen;  how  many  will  remain? 


;4  SIMPLE    SUBTRACTION. 

24.  Talge  eight  from  seventeen ;  how  many  will  remain  i 

25.  Take  nine  from  sixteen;  how  many  will  remain? 

26.  Take  nine  from  eighteen;  how  many  will  remain? 

RULE. 

Place  the  larger  number  uppermost,  and  the  smaller 
one  under  it,  so  that  units  may  stand  under  units;  tens 
under  tens ;  hundreds  ander  hundreds,  &c.  Draw  a  line 
underneath,  and  beginning  with  units,  subtract  the  lower 
4rom  the  upper  figure,  and  set  down  the  remainder. — 
But  when  in  any  place  the  lower  figure  is  larger  than 
the  upper,  call  the  upper  one  ten  more  than  it  really  is; 
subtract  the  lower  figure  from  the  upper,  considering  it 
;is  having  ten  added  to  it,  and  setting  down  the  remain- 
der add  one  to  the  next  left  figure  of  the  lower  line,  and 
proceed  thus  through  the  whole. 
PROOF. 

Add  the  remainder  and  the  less  line  together,  and  if 
the  work  be  right,  their  sum  will  be  equal  to  the  grea- 
ter line. 

EXAMPLES. 
I.  II.  III.  IV.  V. 

23    457    54367    73214    84201 
11    215    20154    54876    49983 


12    242    34213    18338    34218 


VI.  VII. 

98 r 2030405321      700000000000 
6054123456789       98765432123 


APPLICATION. 

1.  A  man  borrowed  of  his  friend  four  hundred  and 
eighty  dollars;  and  having  afterwards  paid  one  hundred 
and  sixty-five,  how  much  was  still  due?      Ans.  315  dolls. 

2.  A  owed  B   10,000  dollars.     He  paid  at  one  time 
467,  and  afterwards  297.     How  much  was  still  due  to 
B?  Ans.  9236  dollars. 


SIMPLE   MULTIPLICATION.  \b 

3.  B  owed  C  11,989  dollars,     lie  paid  atone  time 
2875  dollars;  at  another,  4243;  and  afterwards,  3000. 
How  much  did  he  still  owe?  Ans.  1871  dollars. 

4.  A  man  travelled  till  he  found  himself  1300  miles 
from  home.     On  his  return,  he  travelled  in  one  week 
235  miles;  in  the  next,  275;  in  the  next,  325,  and  in  the 
following  week  290.     How  far  had  h*  still  to  go  before 
he  would  reach  home?  Ans.  175  miles. 

Question   1.  What  does  Subtraction  teach? 

2.  How  do  you  place  the  larger  and  smaller 

numbers? 

3.  What  do  you  do  when  the  lower  number  is' 

larger  than  the  upper  number? 

4.  How  is  a  sum  in  Subtraction  proved? 


SIMPLE  MULTIPLICATION. 

Simple  Multiplication  teaches  a  short  method  of  finding 
what  a  number  amounts  to  when  repeated  a  given  num- 
ber of  times,  and  thus  performs  Addition  in  a  very  ex- 
peditious manner. 

1.  What  will  four  apples  cost,  at  two  cents  a  piece? 

2.  What  must  you  give  for  two  oranges,  at  six  cents 
a  piece? 

3.  What  are  two  barrels  of  flour  worth,  at  five  do,!-^ 
lars  a  barrel? 

4.  What  will  three  pounds  of  butter  come  to,  at  three 
cents  a  pound? 

5.  If  you  can  walk  four  miles  in  one  hour;  how  far 
can  you  walk  in  three  hours? 

6.  If  a  cent  will  buy  five  nuts;  how  many  nuts  wifl 
four  cents  buy? 

7.  What  are   two  barrels  of  cider  worth,  at  three 
dollars  a  barrel? 

Before  entering  upon  this  Rule,  let  the  pupil  so  learn  the  fol- 
lowing table,  as  to  answer  with  readiness  any  question  im 
in  it 4  after  which,  he  will  be  able  to  proceed  with  facility. 


DIMPLE   MULTIPLICATION 


MULTIPLICATION   TABLE. 


Twice 

3  times 

4  times 

5  times 

6  times 

7  times 

Imake2 

1  make3 

1  make  4 

1  make5 

1  make6 

Imake7 

2           4 

2           6 

2          8 

2        10 

2        12 

2        14 

3           6 

3           9 

3        12 

3       15 

3       18 

3       21 

4           8 

4         12 

4        16 

4       20 

4       24 

4       28 

&        10 

5         15 

5       20 

5       25 

5       30 

5       35 

6        12 

6        18 

6       24 

6       30 

6       36 

6       42 

7         14 

7        2*1 

7       28 

7       35 

7       42 

7       49 

8        16 

8        24 

8       32 

8       40 

8       48 

8       56 

r  9     is 

9        27 

9       36 

9       45 

9       54 

9       63 

10        20 

10        30 

10       40 

10       50 

10       60 

10      70 

11        22 

11         33 

11        44 

11        55 

11       66 

11       77 

12        24 

12        36 

12       48 

12       60 

12       72 

12       84 

8  times 

9  times 

10  time? 

1  1  times 

12  times 

1  make  8 

1  make  9 

1  make  10 

1  make  11 

1  make  12 

2    16 

2    18 

2     20 

2     22 

2     24 

3    24 

3    27 

3     30 

3     33 

3     36 

4.    32 

4    36 

4     40 

4     44 

4     48 

'5    40 

5    45 

5     50 

5     55 

5     60 

6    48 

6    54 

6     60 

6     66 

6     72 

7    56 

7    63 

7     70 

7     77 

7     84 

8    64 

8    72 

8     80 

8     88 

8     96 

9    72 

9    81 

9     90 

9     99 

9    108 

10    80 

10    90 

10    100 

10    110 

10    120 

11     88 

11    99 

11    110 

11    121 

11    132 

:12    96 

12   108 

12    120 

12    132 

12    144 

Though  the  foregoing  table  extends  no  farther  than 
12,  it.  may  be  easily  continued  farther;  and  if  pupils 
were  to  extend  it,  and  commit  it  to  memory,  as  far  as  30 
or  40,  it  would  afford  them 'great  advantage  in  their  pro- 
i^rees. 

The  number  to  be  multiplied  is  called  the  multipli- 
cand. 

The  number  which  multiplies  is  called  the  multiplier.* 

The  number  produced  by  the  operation  is  called  the 
product. 


*  The  multiplier  and  multiplicand  arc  also  called  farfa'  •• 


SIMPLE    MULTIPLICATION.  17 

CASE    I» 

When  the  Multiplier  is  no  more  than  12. 
RULE. 

Place  the  greater  number, or  multiplicand, uppermost; 
set  the  multiplier  under  it,  and  beginning  with  units, 
multiply  all  the  figures  of  the  multiplicand  in  succession, 
carrying  one  to  the  next  figure  for  every  ten,  and  setting 
down  the  several  products,  as  in  Addition.  The  whole 
of  the  last  product  must  be  set  down. 

PROOF, 

Multiply  the  sum  by  double  the  amount  of  the  multi- 
plier, and  if  the  work  in  both  instances  be  right,  the  pro- 
duct will  be  double  the  amount  of  the  former  product.* 

EXAMPLES. 
I.  II.  Ill*  IV.  V. 

234    3201    51000   43201    354610 
2345          6 


468   9603  204000  216005  2127660 


VI.  VII.  VIII. 

453210      3245Q13      987654321 
78  9 


3172470     25960104     8888888889 


IX.  X.  XI. 

678987654    321234567    898765432 
9  11  IS 


*  Multiplication  may  be  proved  by  Division;  for  if  the  product  be  divided  by  tjhe 
multiplier,  the  quotient  will  be  the  same  as  the  multiplicand, 

2* 


18 


-SIMPLE    MULTIPLICATION. 


CASE   It. 

When  the  Multiplier  is  more  than  1£. 
RULE. 

Multiply  each  figure  in  the  multiplicand  by  every 
figure  in  the  multiplier,  and  place  the  first  figure  of  each 
product  exactly  under  its  multiplier;  then  a'dd  the  seve- 
ral proddcts  together,  and  their  sum  will  be  the  answer. 

When  cyphers  occur  at  the  right  hand  of  either  of 
the  factors,  omit  them  in  multiplying,  and  annex  them  to 
the  right  hand  of  the  product.* 

When  the  multiplier  is  the  product  of  any  two  whole 
numbers,  the  multiplication  may  be  performed  by  mul- 
tiplying the  sum  by  one  of  them,  and  the  preduct  by  the 
other.  Thus,  if  24  were  to  be  multiplied  by  18,  (as  6 
times  3  make  18.)  let  it  be  multiplied  by  6,  the  product 
by  3,  and  the  answer  will  be  the  same  as  if  multiplied 
by  18. 

EXAMPLES. 


4  3  0  2  1  6  7  ft- 
432 

86043356 
1  29065034 
172086712 

18585364896 

in. 

679100 
32 


13582 
20373 

2  173  1200 


1041 72 
78129 


87654320 
543 

262962960 
3506 1728O 
438271600 

47596295760 

v. 

432000 
4300 


1296 
1728 


885462 


1857600000 


*  Multiplying  by  10,  add  a  cypher  to  tbc  right  hand  sride  of  the  sum,  and  it  is  done- 
Thus,  let  it  be  required  to  multiply  12  by  10,  the  product  will  be  120;  but  if  a  cy- 
pher be  added,  it  will  bring  the  same  result.  In  multiplying  by  100,  add  two  cyV 
phew:  by  1000,  three,  &c. 


SIMPLE   MULTIPLICATION".  19 


Multiply  18450  by  35. 


VI. 

18450 

7 

129150  92250          92250 

5  7         55350 


645750          645750         645750 


9.    multiply    420  by       7  product         2940 

10.  3240  9  29160 

11.  54134  18  974412 

12.  37990  24  911760 

13.  84522  54  4564188 

14.  90203  587  52949161 

15.  370456  7854  2909561424 

16.  7654876  8765  67094988140 

APPLICATION. 

1 .  A  man  had  29  cows,  and  his  neighbour  had  five  times 
as  many.     How  many  had  his  neighbour?         Ans.  145. 

2.  There  are  12  barrels  of  sugar,  each  containing  256 
pounds.     How  many  pounds  did  they  all  contain  ? 

Ans.  3072. 

3.  How  far  will  a  man  travel  in  a  year,  allowing  the 
year  to  contain  365  days,  if  he  travel  40  miles  per  day? 

Ans.  14600  miles. 

4.  In  one  hogshead  are  63  gallons ; — how  many  gallons 

are  there  in  144  hogsheads?  Ans,  9072, 

I 
Q.  1.  What  does  simple  Multiplication  feach? 

2.  What  is  the  number  to  be  multiplied,  called? 

3.  What  is  the  number  called  which  is  used  in  mul- 

tiplying another  number? 

4.  Are  the  multiplicand  and  multiplier  called  by  any 

other  names? 

5.  How  do  you  proceed  when  the  multiplier  is  no 

more  than  12? 

6.  When  the  multiplier  is  more  than  125howdoyou 

proceed? 


20  SIMPLE  DIVISION. 

7.  What  do  you  do  when  cyphers  occur  at  the  right 

hand  of  either  of  the  factors? 

8.  How  do  you  proceed  when  the  multiplier  is  the 

product  of  two  other  numbers? 

9.  How  may  sums  in  Multiplication  be  proved? 


SIMPLE  DIVISION. 

Simple  Division  teaches  to  find  how  often  one  num- 
ber is  contained  in  another,  and  is  a  concise  way  of  per- 
forming several  subtractions. 

Questions  to  prepare  the  learner  for  this  rule. 

1.  James  had  4  apples  and  John  half  as  many;  how 
many  had  John? 

2.  If  two  oranges  cost  6  cents,  what  does  one  cost? 

3.  If  you  divide   8  apples  equally  between  2  boys, 
how  many  will  each  have? 

4.  What  is  one  half  of  eight? 

5.  If  you  divide    6  nuts  equally  among  3  boys,  how 
many  will  each  have? 

6.  What  is  one  third  of  six? 

7.  If  12  cherries  cost  nine  cents,  what  will  4  cost? 

8.  A  third  of  9  is  how  many? 

9.  If  you  divide  16  nuts  equally  among  4  boys,  how 
many  will  each  have? 

10.  A  fourth  of  16  is  how  many? 

11.  How  many  times  two  are  there  in  six? 

12.  How  many  times  three  in  six? 

13.  How  m£hy  times  four  in  eight? 

14.  How  many  times  two  in  twelve? 

15.  In  nine,  how  many  times  three? 

16.  In  eight,  how  many  times  two? 

17.  In  ten,  how  many  times  five? 

18.  In  twelve,  how  many  times  three? 

19.  In  twelve,  how  many  times  four? 

20.  In  twenty,  how  many  times  five? 

21.  In  eighteen,  how  many  times  six? 

22.  In  sixteen,  how  many  times  two? 


SIMPLE   DIVISION.  21 

23.  In  thirty,  how  many  times  five? 

24.  In  thirty,  how  many  times  six? 

25.  In  twenty-one,  how  many  times  seven? 

26.  In  twenty-eight,  how  many  times  seven? 

27.  In  thirty-six,  how  many  times  twelve? 

28.  In  forty-eight,  how  many  times  twelve? 

29.  In  forty-eight,  how  many  times  sixteen? 

30.  In  fifty-five,  how  many  times  eleven? 

31.  In  sixty,  how  many  times  twenty? 

32.  In  eighty,  how  many  times  twenty? 

33.  In  one  hundred,  how  many  times  twenty? 

34.  In  one  hundred  and  twenty,  how  many  times  thirty  ? 

35.  In  ten,  how  many  times  four? 
Answer.     Two  times,  and  two  remain. 

36.  In  fourteen,  how  many  times  three? 
Answer.    Four  times,  and  two  remain. 

37.  In  twenty-five,  how  many  times  four? 
Answer.     Six,  and  one  remains. 

There  are  in  Division  four  principal  parts,  *iz: 
The  dividend,  or  number  to  be  divided. 
The  divisor,  or  number  given  to  divide  by. 
The  quotient,  or  answer,  which  shows  how  many 

times  the  divisor  is  contained  in  the  dividend. 
The  remainder,  which  is  any  overplus  of  figures 

that  may  remain  after  the  sum  is  done,  and  is 

always  less  than  the  divisor. 

CASE   I. 

RULE. — First,  find  how  many  times,  the  divisor  is  con- 
tained in  as^  many  figures  on  the  left  hand  of  the  divi- 
dend as  are  necessary  for  the  operation,  and  place  the 
number  in  the  quotient.  Multiply  the  divisor  by  this 
number,  and  set  the  product  under  the  figures  at  the  left 
hand  of  the  dividend  before  mentioned.  Subtract  this 
product  from  that  part  of  the  dividend  under  which  it 
stands,  and  to  the  remainder  bring  down  the  next  figure 
of  the  dividend;  but  if  this  will  not  contain  the  divisor, 
place  a  cypher  in  the  quotient,  and  bring  down  another 
figure  of  the  dividend,  and  so  on,  until  it  will  contain  the 


22  SIMPLE   DIVISION. 

divisor.  Divide  this  remainder  (thus  increased)  in  the 
same  manner  as  before;  and  proceed  in  this  manner  un- 
til all  the  figures  in  the  dividend  are  brought  down  and 
used. 

PROOF. 

Multiply  the  quotient  by  the  divisor,  and  to  the  pro- 
duct add  the  last  remainder,  if  there  be  any;  if  the 
work  is  right,  the  sum  will  be  equal  to  the  dividend. 

EXAMPLES. 
I. 

Divisor.          Dividend.  Quotient. 

3)1439671  82(47989060 


Proof  143967182 

In  this  example,  I  find 

2  9  that  3,  the  divisor,  can 

2  7  not  be  contained  in  the 

first  figure  of  the  divi- 

2  6  dend ;   therefore  I  take 

24  two  figures,  viz:  14,  and 

inquire  how  often  3  is 

27  contained  therein,  which 

27  I  find  to  be  4  times,  and 

•—  put  4  in  the  quotient. — 

1  8  Then    multiplying    the 

1  8  divisor  by  it,  i  set  tho 

product  under  the   14, 

Remainder.          2  in    the    dividend,    and 

find  by  subtracting  that 

;here  is  a  remainder  of  two.  To  this  2,  I  bring  down 
the  next  figure  in  the  dividend,  viz:  3,  which  increases 
the  remainder  to  23.  I  then  seek  how  often  3  is  con- 
tained in  23,  and  proceed  as  before.  When  I  bring  down 
the  1  that  is  in  the  dividend,  I  find  that  3  can  not  be  con 
tained  in  it,  and  therefore  place  a  cypher  in  the  quotient 
and  bring  down  the  8,  which  makes  18.  Finding  that  3 
is  contained  6  times  in  18,  and  that  there  is  no  remain- 


SIMPLE    DIVISION.  33 

tier,  I  bring  clown  the  2;  but  as  3  can  not  be  contained 
in  it,  I  place  a  cypher  in  the  quotient,  and  let  2  stand  as 
the  last  remainder.  In  proving  the  sum  by  Multiplica- 
tion, the  2  is  added.  This  mode  of  operation  is  called 
LONG  DIVISION. 

ii.  in. 

2  )  3456789  (  1728394  5  )  6789876  (  1357975 

2255 

14   Proof  3456789  17      Pr.    6789876 

14  15 

5  28 

4       _  25 

16  39 

16  35 

7  48 

6  45 

18  37 

18  35 

9  26 

8  25 

1  1 

IV.  V. 

42)9870(235  320)  1 286401 608 1  (40200050 

84  1280 

147  640 

126  640 

210  1608 

210  1600 

8i 


SIMPLE    DIVISION. 


VI. 

12)301203(25100 
24        12 

61  Pr.  301203 
60 

12 
12 

03 


VII. 

15)218760(14584 
15        15 


68 
60 


72920 
1^584 


87   218760 
75 

126 
120 

60 
60 


VIII. 

€48  )  2468098  (  3808 
1944 

5240 
5184 

5698 
5184 

514 


Proof 


3808 
648 

30464 

15232 
22848 
514 

2468098 


1.  Divide    87654  by    58  Quo.    1511  Rem.    16 

2.  456789  679  672  501 

3.  3875642  7898  490  5622 

4.  98765432  1234  80036  1008 
5  12486240  87654  142  39372 
6.  57289761  7569  7569 

Note. — When  there  is  one  cypher,  or  more,  at  the 
right  hand  of  the  divisor,  it  may  be  cut  off;  but  when 
this  is  done,  the  same  number  of  figures  must  be  cutoff 
from  the  right  hand  of  the  dividend;  and  the  figures 
thus  cut  off,  must  be  placed  at  the  right  hand  of  the  re- 
mainder. 


SIMPLE   DIVISION.  25 

EXAMPLES. 
I.  II. 

^100)567434110(94572  18[000)246864|593(13714 

54  18 

27  66 

24  54 

34  128 

30  126 

43  26 

42  18 

14  84 

12  72 

210  Remainder         12593 

Note. — In  dividing  by  10,  100,  or  1000,  &c.  when  you 
cut  off  as  many  figures  from  the  dividend  as  there  are 
cyphers  in  the  divisor,  the  sum  is  done;  for  the  figures 
cut  off  at  the  right  hand  are  the  remainder,  and  those 
at  the  left  are  the  quotient,  as  in  the  following  sums: 

III.  IV. 

Quotient.  Quotient. 

1  |  0  )  9  8  7  6  5  (  4  Rem.  1)00)123456(78  Rem, 

Quo.  Quo. 

1^000)56789(876  Rem.  1  ',0000)8765(4321  Rem. 

CASE   II. 

When  the  divisor  does  not  exceed  12,  seek  how  often 
it  is  contained  in  the  first  figure  or  figures  of  the  divi- 
dend, and  place  the  result  in  the  quotient.  Then  mul- 
tiply in  your  mind  the  divisor  by  the  figure  placed  in  the 
quotient,  subtract  the  product  from  the  figure  under 
which  it  would  properly  stand  in  the  former  case  of  di- 
vision and  conceive  the  remainder,  if  there  be  any,  to 
be  prefixed  to  the  next  figure.  See  how  often  the  divi- 
sor is  contained  in  these,  and  proceed,  as  before,  'till 
3 


26  SIMPLE   DIVISION. 

the  whole  is  divided.     This  operation  is  called  S 
DIVISION. 

EXAMPLES. 

I.  II. 

4)987654321  8)1  2  3  4567189 


Quo.  2469  13580—  I  Quo.    15432098—6 

In  the  first  example,  J  find  that  4  is  contained  twice 
in  9,  and  that  1  remains.  The  1, 1  conceive  as  prefixed 
to  the  next  figure,  which  is  8,  and  they  become  18.  In 
18,  1  find  4  is  contained  four  times,  and  2  remain.  By 
prefixing  the  2  to  the  following  figure,  which  is  7,  they 
make  27.  In  this  manner  I  proceed,  setting  the  result 
of  each  calculation  in  the  lower  line  which  is  the  quo- 
tient. In  the  second  example,  as  8  can  not  be  contained 
in  1,  I  take  two  figures,  and  proceed  as  in  the  first. 

III.  IV. 

9)1023684200         12)1914678987 

v.  vi. 

11)6789870062        12)1000001246 

Note. — When  the  divisor  is  of  such  a  number  that  two 
figures  being  multiplied  together  will  produce  it,  divide 
the  dividend  by  one  of  those  figures,  the  quotient  thence 
arising,  by  the  other  figure,  and  it  will  give  the  quotient 
required.  As  it  sometimes  happens  that  there  is  a  re- 
mainder to  each  of  the  quotients,  and  neither  of  them 
the  true  one,  it  may  be  found  thus: — Multiply  the  first 
divisor  by  the  last  remainder,  and  to  the  product  add 
the  first  remainder,  which  will  give  the  true  one. 

EXAMPLES, 
I. 

Divide  249738  by  56. 
8)249738 

8 

7|31217—2  4 

4459—4         .32       .* 
2 

64  Remainder. 


SIMPLE    DIVISION* 

The  same  done  by  Long  Division, 
56)249738(4459 

224 

257 
224 

333 
280 


538 
504 

3  4  Remainder. 

n. 

Divide  1847562324  by  84. 
42)1847562324    "  7)1847562324 

7)153963527  12)26393747  4—6 

2199478  9—4  2199478  9—6 

12  7 

4  8  Ren*  4  2 

6 

Rem.  4  8 

3.  Divide  84630986  by  72. 

4.  «  6  7  8  6  0  1  2  1  by  63. 

5.  "  1  2  4  5  6  7  4  3  by  96. 

6.  "  34210390  by  81. 

7.  "  54697283  by  1 0  3. 

8.  "  75392618  by  1  1 3. 

Note. — In  all  cases  in  Division,  when  there  is  any  re- 
mainder, the  remainder  and  divisor  form  a  Vulgar 
Fraction.  Thus,  if  the  divisor  be  8  and  the  remainder 
5,  they  make  f  or  five  eights;  or,  as  in  one  of  the  pre- 
ceding examples,  the  divisor  is  56  and  the  remainder  34; 
which  make  ff . 


28  FEDERAL    MONEY. 

APPLICATION. 

1.  If  48672  dollars  be  equally  divided  among  four  song? 
how  much  will  each  receive?  Ans.  12168  dollars, 

2.  If  a  field  of  32  acres,  produces  1 920  bushels  of  corn, 
how  much  is  it  per  acre.  Ans.  60  bushels. 

3.  Sixty  men  at  a  festival,  which  lasted  three  days, 
spent  240  dollars  per  day.     How  much  did  each  man 
spend  per  day,  and  how  much  did  he  spend  in  the  whole? 

Ans.  4  dollars  per  day  and  12  in  the  whole. 

4.  Divide  151200  Ibs.  of  meat,  equally,  among  an  ar- 
my which  consists  of  27  regiments,  each  regiment  ha- 
ving 7  companies,  and  each  company   100  men;    and 
what  would  be  each  man's  share?  Ans.  8  Ibs, 
Q.  1.  What  does  Simple  Division  teach? 

2.  What  are  the  four  principal  parts  of  Division? 

3.  How  do  you  proceed  when  there  is  one  cypher  or 

more  on  the  right  hand  of  the  divisor? 

4.  How  do  you  proceed  in  dividing  by  ten,  or  a  hun- 

dred, or  a  thousand? 

5.  How  do  you  proceed  when  the  divisor  does  not 

exceed  12? 
»).  When  you  divide  by  any  number  not  exceeding 

12,  what  is  the  operation  called? 
7.  When  the  divisor  is  of  such  a  number  that  two 

figures  multiplied  together  will  produce  it? 
3.  What  can  be  vmade  by  placing  the  remainder  of  a 

sum  over  the  divisor?     Ans.  a  VulgaRFraction, 
9.  How  is  a  sum  in  Division  proved? 


FEDERAL  MONEY. 

The  denominations  of  Federal  Money,  or  the  money 
of  the  UMTED  STATES,  are,  Eagle,  Dollar,  Dime,  Cent, 
>.nd  Mill. 

TABLE, 

10  Mills  (in)  make  1  Cent,  c. 

10  Cents      -  1  Dime,  d. 

10  Dimes    -  1  Dollar,  /).  or  g. 

10  Dollars  -  1  Eagle,  E. 


29 

,  fn  writing  Federal  Money,  it  is  ctistomary  to  omit  Ea- 
gles, Dimes,  and  Mills,  and  set  down  sums  in  dollars, 
cents,  and  parts  of  a  cent.  The  parts  of  a  cent  gene- 
rally used  are,  halves,  thirds,  and  quarters.  Thus,  \  is 
a  half;  -J  a  third;  \  a  quarter.* 

As  the  column  of  cents  admits  of  any  number  under 
one  hundred,  it  consists  of  two  rows  of  figures ;  and  when 
a  less  number  than  10  is  written,  a  cypher  is  placed  to 
the  left  hand  of  it.  In  writing  a  sum  in  dollars  and 
cents,  if  any  part  of  it  consist  of  even  dollars  without 
cents,  the  place  of  cents  is  supplied  with  two  cyphers, 
Cents  are  separated  from  dollars  by  a  point  or  period* 
Exercises  for  the  learner. 

1.  How  many  mills  make  a  cent? — How  many  half  a 
cent? — How  many  a  cent  and  a  half? — How  many  twp 
cents? 

2.  How  many  halves  of  a  cent  make  one  cent? 

3.  How  many  thirds  of  a  cent  make  a  cent? 

4.  How  many  fourths  of  a  cent  make  a  half  cent? 

5.  How  many  fourths  make  a  cent? 

6.  How  many  cents  make  one  fourth  or  quarter  of  a 

dollar. 
*7.  How  many  cents  make  a  half  dollar? 

8.  How  many  cents  make  three-fourths  of  a  dollar? 

9.  How  many  cents  make  a  dollar? 

10.  How  many  dollars  and  cents  in  one  hundred  and 
ten  cents? — How  many  in  two  hundred  and  six  cents? — 
How  many  in  three  hundred  and  forty-eight  cents? — 
How  many  in  five  hundred  and  one  cents? 

11.  If  you  give  a  dollar  for  a  book,  thirty  cents  for  a 
slate,  and  one  cent  for  a  pencil;  how  many  cents  will 
you  give  for  the  whole? 

12.  Write  down  one  dollar  and  eight  c«nts.     Two  dol- 
lars a»d  sixteen  cents.     Twenty  dollars  and  five  cents? 

13.  Write  down  three  hundred  dollars  and  forty  cents, 

14.  Five  hundred  eighty-four  dollars  and  fifty  cents. 

*  In  addition,  subtraction,  and  division  of  Federal  Money,  the  partp  of  a  cent 
less  than  a  fourth  are  usually  omitted.  A  part  greater  than  a  fourth  jfe  Called  fr 
half,  or  three-fourths,  according  to  its  proportionate  value. 

3* 


30  FEDERAL   MONEY. 

15.  Eight  hundred  sixty  dollars  and  sixty-seven  cents, 

16.  Four  thousand  eight  hundred  dollars  and  two  cents. 

17.  Six  hundred  thirty-one  dollars  fifty-six  and  a  fourth 
cents. 

18.  Nine  hundred  and  eighty-seven  dollars. 

19.  Thirty-two  thousand  five  hundred  dollars  eighty- 
seven  and  a  half  cents. 

20.  Ten  dollars  sixty-eight  and  three-fourths  cents. 

21.  Twelve  dollars  ninety-three  and  three-fourths  cents, 

22.  Twenty  dollars  thirty-seven  and  a  half  cents. 

23.  If hirty- three  dollars  thirty-three  and  a  third  cents. 

24.  Sixty  dollars  sixty-six  and  two  third  cents. 

25.  Read  the  following  sums,  viz: 

$8448.871    $3450.25   $47967.91    $7.10   $115.331 
$170.93$        $19.01       $85.061 


ADDITION   OF   FEDERAL   MONEY. 

RULE. 

Begin  at  the  right  hand  side  of  the  sum,  add  one  row 
of  figures  at  a  time,  and  carry  one  for  every  ten,  from 
the  lower  denomination  to  the  next  higher,  as  in  Simple 
Addition,  until  the  whole  is  added.  When  you  come  to 
the  last  row  on  the  left  hand,  instead  of  setting  down 
what  remains  over  ten,  twenty,  or  thirty,  £,c.  set  down 
the  full  amount. 

Note. — When  there  are  parts  of  a  cent  in  a  sum,  such 
as  halves,  &c.  find  the  amount  of  them  in  fourths  of  a 
cent;  consider  how  many  cents  these  fourths  will  make, 
and  add  them  to  the  first  row  in  the  column  of  cents. — 
When  the  parts  of  a  cent  are  not  sufficient  to  make  a 
cent,  place  their  amount  at  the  right  hand  of  the  column 
of  cents, as  in  the  first  example;  and  when  the  parts  of 
a  cent  make  one  cent  or  more,  and  some  parts  remain, 
but  not  enough  for  another  cent,  the  parts  thus  remaining 
must  be  set  down  in  the  same  way,  according  to  the  se- 
cond example.  The  proof  is  the  same  as  in  Simple  Ad- 
dition, 


FEDERAL    MONEY.  %                31 
EXAMPLES. 

j.           ii.        in.  iv 

D.  cts.     D.  cts.     D.  cts.  D.   cis. 

5432,121     324.871     885.90  98765432 

1234.561     987. 43f     125.871  123456.78 

7898.76      720.30     440.40  987654.32 

5432.12      842. 43J     867.121  123000.45 

3456.78      100.621    390.97  678987.65 


23454 . 34|    2975 . 671    2710.27    2900753 . 52 


APPLICATION. 

1.  A  man  bought  a  farm  in  five  parcels;  for  the  first, 
lie  gave  $250.75;  for  the  second,  $350;  for  the  third, 
$475.871;  for  the  fourth,  $550;  and  for  the  fifth,  $600. 
What  was  paid  for  the  farm?  Ans.  2226.621 

2.  A  merchant,  in  buying,  gave  for  flour,  $325.43£; 
for  sugar,  $854.25;  for  molasses,  $520.621;  for  coffee, 
$944.50;  and  for  cotton,  $6427.121.     What  was  the  sum 
paid?  Ans.  $9071, 93|. 

3.  What  is  the  amount  of  101  cents;   93|  cents;  871 
cents;  50  cts.;  311  cts.;  43|  cts.;  and  11  dollars? 

Ans.  $14.16f  cents. 

4.  Gave  for  an  Arithmetick  31i  cents;  fora  slate,  371 
cents;  for  quills,  50  cents;  for  an  inkstand,  621  cents; 
for  a  Geography,  1  dollar,  and  for  a  History,  87i  cents. 
How  much  do  they  amount  to?  Ans.  $3.  68|  cts. 

5.  Add  $75212.50.  $90000,  $644225.75, 

$4587220.50,  and   $5876432.75. 


SUBTRACTION   OF   FEDERAL   MONEY, 

RULE. 

Place  the  smaller  sum  under  the  larger,  setting  the 
dollars  under  dollars  and  cents  under  cents,  and  proceed 
as  in  Simple  Subtraction.  When  there  is  a  fraction,  or 
part  of  a  cent  in  the  upper  line  of  figures,  and  none  m 
the  lower,  set  it  down  at  the  right  of  the  remainder,  as 


52  FEDERAL    MONfcY, 

a  part  of  the  answer.  When  there  is  a  fraction  in  each 
line,  and  the  upper  one  is  the  larger,  subtract  the  lower 
©ne  from  it  and  set  down  the  difference ;  but  if  the  lower 
one  is  larger  than  the  upper,  subtract  it  from  the  nunv 
ber  that  it  takes  of  the  fraction  to  make  a  cent — add 
the  difference  to  the  upper  one,  and  set  down  the  amount. 
When  there  is  a  fraction  in  the  lower  line  and  none  in 
the  upper;  subtract  the  fraction  from  the  number  that 
it  takes  of  it  to  make  a  cent,  and  set  down  the  remain- 
der. In  this  case,  and  likewise  when  the  part  or  frac- 
tion below  is  larger  than  the  upper  one,  it  is  necessary 
to  carry  one  to  the  right  hand  figure  of  the  lower  row 
of  cents. 

EXAMPLES. 

lit.  IV, 

D.  c.  D.  c. 
687.31  9000.43 
599.81  8220.3U 


$294.75      $87.50      $780.1 1| 

VII.  VIII. 

D.     c.  D.        c. 

5978.311         9810000.12| 
4689.93$         1987654.68| 

$237.75  $30.061         $1288.371      $7822345.441 

9.  Subtract  $987.20  from  $1000. 

10.  Subtract  $5871.311,  from  $6430.87J. 

11.  Take  $44.87i,  from  300  dollars. 

12.  Take  $11000,  from  $19876.871. 

APPLICATION. 

1.  Bought  goods  amounting  to  $4875.621,  and  having 
paid  $2850.93$;  how  much  remains  due? 

Ans.  $2*024.68$. 

2.  My  account  against  my  neighbour  amounts  to  $759. 
55;  and  his  account  against  me  js  $546. 87-J.    How  much 
does  he  owe  me?  Ans,  $2 12,37  j. 


FEDERAL    MONEY.  33 

3.  Having  bought  a  quantity  of  goods  at  $5425,  and 
sold  them  at  $6932.68£.     How  much  did  I  make  on  the 
goods?  Ans.  $1507.68f 

4.  A  owes  me  $  11587. 50,  but  having  failed  in  busi- 
ness, he  is  able  to  pay  $9263.621.     How  much  do  I  lose? 

Ans.  $2323.87-1, 

5.  Subtract  $8427.871,  from  $9000.     Ans.  $572. 12-J, 


MULTIPLICATION  OF  FEDERAL  MONEY. 

RULE. 

Set  the  multiplier  under  the  sum,  and  proceed  as  in 
Simple  Multiplication,  carrying  one  for  every  ten  from 
a  lower  to  a  higher  denomination,  until  Ihe  whole  is 
multiplied.  After  the  sum  is  done,  separate,  by  a  pe- 
riod, the  two  right  hand  figures  of  the  product  for  cents, 
and  the  figures  at  the  left  hand  of  the  period  will  be 
dollars. 

Note. — When  the  sum  to  be  multiplied  contains  a  frac- 
tion, or  part  of  a  cent,  multiply  it  by  the  multiplier,  and 
consider  how  many  cents  are  contained  in  its  product. — 
Then  multiply  the  first  figure  of  the  cents  and  add  to 
its  product  the  cents  contained  in  the  product  of  the 
fraction,  and  proceed  as  directed  above.  In  multiplying1 
a  fraction,  if  you  find  in  the  product  one  cent  or  more, 
and  a  remainder  not  large  enough  to  make  another  cent,, 
set  it  down  at  the  right  hand  of  the  product,  that  is  un- 
der the  row  of  fractions  or  parts  of  a  cent.  When  there 
is  a  fraction  in  jthe  sum,  and  the  multiplier  exceeds  12, 
multiply  the  sum  without  the  fraction,  and  afterwards 
multiply  the  fraction  and  add  it  to  the  sum. 

EXAMPLES, 

I.  II. 

D.  c.  D.   c. 

124.10    830.121 


248.20   2490.371    689.20  12258.121   3305.25 


FEDERAL  MONEY* 


VI. 
D.     C. 

164325.11 
18 

1314600.88 
1643251.1 

2957851.98 


12480.68 
62403.4 

74884.08 


1542.17 
6609.3 
9* 

8151.561 


9. 
10. 
11. 
12. 
13. 
14. 

16. 
17. 

18. 
19. 
20. 
21. 


Multiply 


$420.50 

by 

2. 

Ans. 

$  84  1.00 

519.75 

by 

3. 

Ans. 

$1559.25 

99.62J 

by 

4. 

Ans. 

398.50 

75.31| 

by 

5. 

Ans. 

376  56| 

62.121 

by 

6. 

Ans. 

372.75 

750.25" 

by 

7. 

Ans. 

5251.75 

330.124 

by 

8. 

Ans. 

2641.00 

248.87| 

by 

9. 

Ans. 

2239.87^ 

95.93J 

by 

12. 

Ans. 

1151.25 

24.17 

by 

10. 

Ans. 

241.70* 

37.50 

by 

28. 

Ans. 

1050.00 

58.93| 

by 

36. 

Ans. 

2121  75 

9876.621  by  208.     Ans.  2054338,00 


APPLICATION. 


1.  How  much  will  18  barrels  of  flour  cost,  at  3  dol- 
lars per  barrel?  Ans.  54  dollars, 

2.  What  will  35  pounds  of  coffee  cost,  at  20  cents  per 
pound?  Ans  7  dollars. 

3.  Sold  87  barrels  of  flour,  at   $3.121  per  barrel. 
What  was  the  amount?  Ans.  $271.871. 

4.  Bought    160  acres  of  land,  at  $1.25  per  acre.  — 
What  did  the  whole  cost?  Ans.  20Q  dollars. 

5.  What  will  225  bushels  of  apples  cost,  at  62i  cents 
per  bushel?  Ans.  140.621. 

6.  What  will  580  bushels  of  salt  cost,  at  $1.12}  per 
bushel?  Ans.  $652.50 

*  In  multiplying  by  10,  when  there  is  no  fraction  in  the  sum,  it  is  necessary  to 
add  a  cypher  to  tho  right  hand  of  the  mm,  placing  the  period  that  separates  cents, 
t'rom  dollars  one  figure  farther  towards  the  ripht  hand,  and  the  gurn  is  done.  In 
«nu!tiplying  by  100,  add  two  cyphers;  l>y  1000,  three,  &c. 


FEDERAL   MONEY.  35 

DIVISION   OF   FEDERAL   MONEY. 

RULE. 

Proceed  as  in  Simple  Division.  When  the  sum  con 
rsists  of  dollars  and  cents,  the  two  right  hand  figures  of 
the  quotient  will  be  cents.  When  there  is  a  remainder, 
multiply  it  by  4,  adding  the  number  of  fourths  that  are 
in  the  fraction  of  the  sum  (if  there  be  any)  to  its  pro- 
duct: then  divide  this  product  by  the  divisor,  and  its 
quotient  will  be  fourths,  which  must  be  annexed  to  the 
quotient  of  the  sum.  When  the  sum  consists  of  dollars 
only,  if  there  be  a  remainder,  add  two  cyphers  to  it; 
then  divide  by  the  divisor  as  before,  and  its  quotient 
will  be  cents,  which  must  be  added  to  the  quotient  of 
the  sum.  When  the  sum  is  in  dollars,  and  the  divisor  is 
larger  than  the  dividend,  add  two  cyphers  to  the  divi- 
dend— then  divide,  and  the  quotient  will  be  in  cents, 

EXAMPLES. 
I.  II.  III.  IV. 

D.  v.      D.  c.      D.  c.      D.  c.  D.  c, 
2)420.50    4)8000.00    5)580.75    7)84.49(12.67 


7 
210.25     2000.00     116.15 

14 
14 


49 
49 


V.                    VI.  VII. 

D.  c.  D.  c.       D.  c.  &:  c.  D.  c.  D.  c 
9)27.81(3.09     12)144.60(12.05     36)162.36(4.5! 

27             12  144 

81            24  183 

81            24  180 

60  36 

60  36  * 


36  FEDERAL  MONE& 

VIII.  IX. 

D.   c.  D.  c.  D.   c.   D.  c. 

36)1234.72(34.29£--          44)87654.3*2(1992.  ?4i 
108  44 

436 
396 


107  405 

72  396 

352  94 

324  88 

28  63 

4  44 

36)112(3  192 

108  176 

4  16 

4 

44)64(1 
44 

20-f 


10. 

Divide 

£640 

by 

12 

11. 

u 

717.  12£ 

by 

8 

12. 

a 

246.25 

by 

9 

13. 

u 

687  .  20 

by 

12 

14. 

U 

980 

by 

34 

15. 

u 

87654 

by 

128 

16. 

u 

1284  ,31£ 

by 

112 

17. 

u 

40000 

by 

188 

18. 

a 

976  .  871 

by 

225 

19. 

u 

1234.37JL 

•by 

212 

SO. 

u 

9876  .  44 

by 

345 

21. 

a 

89876  .  54 

by 

374 

FEDERAL    MONEY.  37 

APPLICATION. 

1.  Divide  400  dollars,  equally,  among  20  persons. — 
What  will  be  the  portion  of  each  person?        Ans.  $20. 

2.  Divide  1728  dollars,  equally,  among  12  persons. 
What  does  each  one  of  them  share?  Ans.  $144. 

3.  If  240  bushe's  cost  420  dollars,-  what  is  the  cost  of 
one  bushel  at  the  same  rate?  Ans.  $1.75. 

Promiscuous  Examples. 

1.  What  will  the  following  sums  amount  to,  when  ad- 
ded together,  namely: — 

$124.621;     $248.871;     $342.40;     $9850.25. 
and  $20.311?  Ans.  $10586.461. 

2.  If  my  estate  is  worth   12870  dollars,  and  I  meet 
with  losses  amounting  to  $4364.50,  how  much  shall  I 
have  left?  Ans.  $8505.50. 

3.  A  merchant  enters  into  a  trade  by  which  he  re- 
ceives $1324.621  per  year,  for  four  years;  how  much  is 
his  whole  gain?  Ans.  $5298.50. 

4.  An  estate  of  98740  dollars  is  to  be  divided,  equally, 
between  8  heirs;  what  did  each  receive? 

Ans.  $12342.50. 
A  bought  of  B, 

1  barrel  of  sugar  at  -  -       $24.50 

1  chest  of  tea,  60.00 

1  hogshead  of  salt,  -         •  3.75 

20  yards  of  cloth,  -  15.00 

1  barrel  of  flour,  -  3.87J 

Ans.  $107.121. 

Q.  1.  What  are  the  denominations  of  Federal  Money? 

2.  How  many  mills  make  a  cent? 

3.  How  many  cents  make  a  dime? 

4.  How  many  dimes  make  a  dollar? 

5.  How  many  dollars  make  an  eagle? 

6.  How  are  the  denominations   generally  used  in 

writing  Federal  Money,  and  in  reckoning? 

7.  Where  is  Federal  Money  used  as  a  currency? 
Answer.     In  the  United  States  of  North  America, 

4 


TABLE 

OF 

MONEY,  WEIGHTS,  MEASURES, 


ENGLISH  MONEY. 

A  table  of  Federal  Money  has  already  been  given. 

The  denominations  of  English  Money  are  pound,  shil 
ling,  penny,  and  farthing. 

4  farthings  (qr.)  make  1  penny         d. 

12  pence       -  1  shilling       s. 

20  shillings  -  1  pound         Jg, 

0^- Farthings  are  written  as  fractions,  thus: 

I-  one  farthing. 

A  two  farthings,  or  a  half-penny. 

J  three  farthings. 


PENCE  TABLE. 

1 

d. 

«.  d. 

s. 

20  pence  make 

1  8 

20 

30  "    "   - 

^  6 

30 

40  "    " 

3  4 

40 

50   «      "    r 

4  2 

50 

60  "    "   - 

5  0 

60 

70  "    " 

5  10 

70 

80  «    «   - 

6  8 

80 

90  «    "   - 

7  6 

9Q 

100  «    «   - 

8  4 

100 

110  «    «-   - 

9  2 

no 

120  «    «   - 

10  0 

120 

240  <«    "   - 

20  0 

130 

SHILLING  TABLE. 

S. 

£.  * 

20    -    -    - 

1  0 

30   -   -   - 

1  10 

40   -   -   - 

2  0 

50   -   -   - 

2  10 

60   -   -   - 

3  0 

•70   -   -   - 

3  10 

80   -  >   - 

4  0 

90   -   -   - 

100 

4  10 
5  0 

no  - 

5  10 

120   -   -   - 

6  0 

130   -   -   - 

6  10 

TABLE    OF    WEIGHTS   AND    MEASURES.,  39 

TROY    WEIGHT. 

By  this  weight,  jewels,  gold,  silver,  and  liquors  are 
Weighed. 

The  denominations  of  Troy  Weight  are  pound,  ounce, 
pennyweight,  and  grain. 

24  grains  (gr.)         make          1  pennyweight     dwt. 

20  pennyweights   -  1  ounce       -         oz. 

12  ounces  1  pound  Ib. 

AVOIRDUPOIS    WEIGHT. 

By  this  weight  are  weighed  things  of  a  coarse,  dros- 
sy nature,  that  are  bought  and  sold  by  weight;  and  all 
metals  but  silver  and  gold. 

The  denominations  of  Avoirdupois  Weight  are  ton, 
hundred  weight,  quarter,  pound,  ounce,  and  dram. 

16  drams,  (dr.)       make       1  ounce  -  oz. 

16  ounces   -  1  pound  -  Ib. 

28  pounds  -  1  quarter  of  a  cwt.  qr. 

4  quarters,  or  112  Ib.          1  hundred-weight    cotf, 

20  hundred  weight    -          1  ton  T. 

APOTHECARIES   WEieHT. 

By  this  weight  apothecaries  mix  their  medicines,  but 
buy  and  sell  by  Avoirdupois  Weight. 

The  denominations  of  Apothecaries  Weight  are 
pound,  ounce,  dram,  scruple,  and  grain. 

20  grains  (gr.)  make  1  scruple         9 

3  scruples    -  1  dram  3 

8  drams       -  1  ounce  3 

12  ounces  1  pound  ft 

LONG   MEASURE. 

Long  measure  is  used  for  lengths  and  distances. 
The   denominations  of  Long  Measure  are  degree, 
league,  mile,  furlong,  pole,  yard,  fcot,  and  inch, 


40  TABLE   OP   MEASURES. 

12  inches  (in.)      make  1  foot                         ft. 

3  feet       -  1  yard  -                  yd. 

5±  yards,  or  16J  feet  1  rod,  pole,  or  perch  P. 

40  poles  (or  220  yds.)  1  furlong  -                 fur. 

8  furlongs  (or  1760  yds.)  1  mile  -  M. 

3  miles    -  1  league  -                  L. 

eOgeographick    >     n  d                       , 
or  69J  statute  3 

Note. — A  hand  is  a  measure  of  4  inches,  and  used  in 
measuring  the  height  of  horses. 

A  fathom  is  6  feet,  and  used  chiefly  in  measuring  the 
depth  of  water. 

CUBICK,   OR   SOLID   MEASURE. 

By  Cubick,  or  Solid  Measure,  are  measured  all  (hings 
that  have  length,  breadth  and  thickness. 

Its  denominations  are,  inches,  feet,  ton,  or  load,  and 
cord. 

1728  inches  make  1  cubick  foot, 

27  feet      -  1  yard. 

40  feet  of  round  timber) 

or  50  feet  of  hewn£  1  ton  or  load, 

timber  } 

128  solid   feet,  i.  e.  8  in) 

length,  4  in  breadth,  >  1  cord  of  wood, 

and  4  in  height        ) 

LAND,   OR   SQUARE    MEASURE. 

This  measure  shows  the  quantity  of  lands. 
The  denominations  of  Land  Measure  are  acre,  rood, 
square  perch,  square  yard,  and  square  foot. 


144  square  inches       make 

9  square  feet 
rJO-J-  square  yards 
40  square  perches 

4  roods 
'.MO  acres 


square  foot  ft. 

square  yard  yd. 

square  perch  1J. 

rood  R. 

acre  A. 

mile          -  m. 


TABLE   OF   MEASURES.  41 

CLOTH   MEASURE. 

By  this  measure  cloth,  tapes,  &c.  are  measured. 
The  denominations  of  Cloth  Measure  are  English  el% 
Flemish  ell,  yard,  quaiter  of  a  yard,  and  nail. 

4  nails  (na)         make         1  quarter  of  a  yard  qr. 

4  quarters  1  yard      -  yd. 

3  quarters  1  ell  Flemish    -         E.  FL 

5  quarters          -  1  ell  English     -         E.  E, 

6  quarters  1  ell  French     -        E.  F, 

DRY   MEASURE. 

This  measure  is  used  for  grain,  fruit,  salt,  &c. 
The  denominations  of  Dry  Measure  are  bushel,  peck7 
quart,  and  pint. 

2  pints  (pt.)         make         1  quart    -  qt. 

8  quarts     -  1  peck  pe, 

4  pecks  1  bushel  bit,,- 

WINE   MEASURE. 

By  Wine  Measure  are  measured  Rum,  Brandy,  Perry., 
Cider,  Mead,  Vinegar  and  Oil. 

Its  denominations  are  pints,  quarts,  gallons,  hogsheads^ 
pipes,  &c. 

2  pints  (pt.)        make         1  quart  qt. 

4  quarts     -  1  gallon  gal, 

42  gallons    -  1  tierce       -         tier. 

63  gallons    -  1  hogshead          hhd. 

2  hogsheads       -  1  pipe  or  butt      P.  or  £« 

2  pipes      -  1  tun  -  T. 

ALE,  OR  BEER  MEASURE* 

The  deneminations  of  this  measure  are  pints,  quartSj 
gallons,  barrels,  &c, 

4* 


12  TABLE   OF    TIME    AND    MOTION. 

2  pints  (pt.)         make  1  quart    -  qls. 

4  quarts  1  gallon  -  gal. 

8  gallons   -  1  firkin  of  ale  -        fir. 

2  firkins     -  1  kilderkin        -         kil. 

2  kilderkins       -  1  barrel    -  bar, 

11  barrels,  or  54  gallons  1  hogshead  of  beer  hhd. 

2  barrels  -  1  puncheon       -        pun. 

3  barrels,  or  2  hogsheads  1  butt  butt. 

TIME. 

The  denominations  of  Time  are  year,  month,  week^ 
day,  hour,  minute,  and  second. 

60  seconds  (sec.)         make         1  minute  -  mm. 

60  minutes        -  1  hour      -  H. 

24  hours  1  day        -  D. 

7  days  1  week  W. 

-52  weeks,  1  day,  atid  6  hours  J   1  v 
or  365  days,  and  6  hours  \   i  ye; 

12  months  (mo.)  1  year 

Note.  —  The  six  hours  in  each  year  are  not  reckoned 
till  they  amount  to  one  day:  hence,  a  common  year  con- 
sists of  365  days,  and  every  fourth  year,  called  leap 
year,  of  366  days. 

The  following  is  a  statement  of  the  number  of  days 
in  each  of  the  twelve  months,  as  they  stand  in  the  calen- 
dar or  almanack: 

The  fourth,  eleventh,  ninth,  and  sixth, 

Have  thirty  days  to  each  affix'd  : 

And  every  other  thirty-one, 

Except  the  second  month  alone, 

Which  has  but  twenty  eight  in  fine, 

Till  leap  year  gives  it  twenty-nine. 

MOTION. 

60  seconds  make     1  prime  minute,  " 

60  minutes      -  1  degree  -  ° 

30  degrees     -  1  sign       -  $. 


12  signs,  or  360  degrees  j        Thef  "*o}%  great  circle 
$  of  the  Zodiack, 


COMPOUND  ADDITION. 


Compound  Addition  teaches  to  add  numbers  which  re 
present  articles  of  different  value,  as  pounds,  shillings, 
pence;  or  yards,  feet,  inches,  &c.  called  different  de- 
nominations. The  operations  are  to  be  regulated  b>  the 
value  of  the  articles,  which  must  be  learned  from  the 
foregoing  table. 

RULE. 

Place  the  numbers  to  be  added  so  that  those  of  the 
same  denomination  may  stand  directly  under  each  other. 
Add  the  figures  of  the  first  column  or  denomination  to- 
gether, and  divide  the  amount  by  the  number  which  it 
takes  of  this  denomination  to  make  one  of  the  next  higher. 
Set  down  the  remainder,  and  carry  the  quotient  to  the 
next  denomination.  Find  the  sum  of  the  next  column 
or  denomination,  and  proceed  as  before  through  the 
whole,  until  you  come  to  the  last  column,  which  must 
be  added  by  carrying  one  for  every  ten  as  in  Simple  Ad- 
dition. 

EXAMPLES. 
I.  II.  III. 

£  s.  d.  qrs.  £  s.  d.  qrs.  £  s.  d.  qrs. 

14  10  8  2  19  19  11  3  18  17  11  3 

11  16  10  1  10  14  4  1  15  J4  10  3 

8  3  11  3  13  13  10  2  17  18  9  2 


34  11  6  2    44   8   2  2    62  11  8  0  Ans, 


In  the  first  of  the  above  examples,  I  begin  with  the 
right  hand  column,  or  that  of  farthings;  and  having  ad- 
ded it,  find  that  it  contains  6.  Now,  as  6  farthings  con 
tain  1  penny  and  2  over,  I  set  the  2  farthings,  under  the 
column  of  farthings,  and  carry  the  penny  to  the  column 
of  pence.  In  the  column  of  pence  I  find  29,  which,  with 


44  COMPOUND    ADDITION. 

the  one  carried  from  the  forth  ings,  make  30.   In  30  pence 

1  find  there  are  2  shillings  and  6  pence  over:  setting  the 
6  pence  under  the  column  of  pence,  I  add  the  2  shillings 
to  the  column  of  shillings.    In  this  column  are  29,  and  the 

2  added  make  31.     Thirty-one  shillings  contain  1  pound, 
and  11  shillings  over.     The  11  shillings  are  then  placed 
under  the  column  of  shillings,  and  the  1  is  carried  to  the 
column  of  pounds.    In  that  column  are  33  pounds,  which, 
with  the  1  added,  make  34.     Thus  the  amount  of  the 
sum  is,  34  pounds,  11  shillings,  6  pence,  and  2  farthings; 

In  all  cases  in  Compound  Addition,  one  must  be  car- 
ried for  the  number  of  times  that  the  higher  denomina- 
tion is  contained  in  the  column  of  the  lower  denomina- 
tion. Thus,  in  Troy  Weight:  as  24  grains  make  one 
pennyweight,  one  from  the  column  of  grains  is  carried 
for  every  24;  in  the  column  of  pennyweights,  one  for 
every  20;  and  in  every  instance  the  learner  must  be 
guided  by  the  foregoing  table  of  "Money,  Weight?, 
Measures,  &c." 

IV.  V. 

£  s.  d.  qrs.  £  s.      d.    qrs. 

487  16  11     3  9876  15       4     5 

830  10  9     1  2123  14       5     0 

500  11  42  6789  18     10     2 

620  18  3     3  1234  15     11      1 

900  8  10     0  7876       493 


,/Vote. — Sums  in  Compound  Addition  njay  be  proved  in 
the  same  manner  as  in  Simple  Addition. 

TROY   WEIGHT. 

VI.  VII. 

Ib.  02.  dwt.  gr.  Ib.  oz.  dwt.  gr. 

487  10  18  22  6780  11   11  12 

500   8  11  10  1100  9  18  22 

234  11   10  16  3090  10  17  20 

876   3  17  23  2468  8  13  19 


COMPOUND    ADDITION.  45 

AVOIRDUPOIS   WEIGHT. 

VIII.  IX. 

Ton.  cwt.  qr.  Ib.     oz.    dr.  Ton.  cwt.  qr.  Ib.   Qz. 

16     18     2     25     11      14              27     17  3  27     8 

97     12     3     17       9     11              98     19  2  11     9 

34     11      1      10     10     10             70     11  1  18     7 

82     19     2     27     15     13              18     16  0  10     6 


APOTHECARIES    WEIGHT. 

X.  XI, 

fc      3    3  6  gr.  ft  3  3  6  £T- 

74       9     7  1  13  20  10  7  1  18 

18     11     6  2  17  37  11  5  2  17 

91      10     3  0  10  28  9  3  1  15 

17       9     5  1  19  14  ft  4  0  11 


LONG  MEASURE. 

XII.  XIII. 

deg.  mil.  fur. po.  ft.  in.  mil.  fur.  po.  yd.  ft. 

118  36  7  19  13  3  976  2  13  4  2 

921  15  4  16  10  10  867  6  10  3  I 

671   10  6  27  11  11  500  1   11  0  0 

643  26  5  15  8  8  123  4  15  3  2 

123  14  5  16  7  8  345  6  17  1  0 


CUBICK,  OR  SOLID  MEASURE. 

XIV.  XV.  XVI. 

Ton  ft.   in.  yd.  ft.  in.  Cord  ft.  in. 

17  10  1229,  29  20  1092  48  120  1630 
24  13  1460  11  11  1195  54  110  1500 
98  25   1527  18  11  1000  75  88  1264 

18  16   1079  27  9  1330  87  113  1128 


46 


COMPOUND    ADDITION. 


LAND>   OR    SQUARE    MEASURE. 


XV II. 

acr.  roo.  per. 

987  2  23 

798  3  28 

123  2  11 

567   1  27 

700  0  00 


xx. 

yd.  qr.  nL 

175  3  3 

481  2  1 

234  1  2 

345  0  1 

234   1  2 


XXIII. 

El.  E.  qr.  nl 

87654   1  2 

56788  3  1 

87654  3  2 

12345  0  0 

84231  2  3 


xxvr. 

bush.  pk.  qt. 

187  7  3 

290  6  2 

185  3  1 

.549   1  2 

160  5  3 


XVIII. 

XIX. 

acr.  roo.  per. 
8423  1  36 

acr.  roo.  per, 
9432  3  24 

1234  0  10 

4324  2  12 

4821  3  11 

5678  1  36 

6789  2  30 

5865  3  11 

8000  1   13 

8765  2  15 

CLOTH  MEASURE 

XXI. 

XXII. 

El.  Fr.  qr.  nl. 

247  2  3 

£J.  Fl.  qr.  nL 
9876  2  3 

456  1   1 

8765  1  2 

345  3  0 

3456  2  3 

236  2  2 

4000  0  0 

567  0   1 

7898  2  3 

XXIV. 

XXV. 

yd.   qr.  nl. 
656547  1   1 

yd.    qr. 
987654321  3 

nl. 
3 

987654  2  0 

234567876  0 

0 

765432  1  3 

543212345  3 

2 

134545  3  2 

900087654  1 

3 

584050  0  1 

384563200  3 

0 

DRY  MEASURE. 

XXVII. 

XXVIII. 

busk.  pk.  qt. 
356  3  7 

bush.  pk.  qt. 
874  3  6 

120  1  6 

123  1  2 

543  2  1 

345  2  5 

678  3  5 

753  1  7 

432  1  :3 

936  2  4 

COMPOUND    ADDITION. 


-47 


WINE    MEASURE. 


XXIX. 

Tun.  hhd.  gal.  qt.  pt. 

4820  1   16  2  1 

9765  3  18  3  1 

8645  2  19  1  0 

5432  1  22  3  1 

6787  1   10  1  0 


XXX. 

Tun.     hhd.  gal.  qt.  pt. 

987654      1      12  1  1 

321234     3     15  0  1 

125780     2     18  3  1 

876531     2     27  1  0 

248765     1     49  2  1 


ALE   OR   BEER   MEASURE. 


XXXI. 

hhd.  gal.  qt.  pt. 

4820  48  3  1 

8765  34  1  1 

9877  53  2  1 

1234  12  1  0 

5678  50  0  1 


XXXII. 

hhd.   gal.  qt.  pt, 

17819174  18  3  1 

21350000  27  1  1 

12168400  35  0  0 

21346870  15  3  1 

43212345  50  1  1 


TIME. 


xxxni. 

flu.  d,  h.  m.  s. 

3  6  23  58  24 

3  5  20  49  57 

1  4  21  30  30 

3  2  13  63  53 

1  0  10  10  10 


XXXIV. 


y.  mo.  'os.  d.  h.  m.  $. 

75  11  3  6  22  50  57 

18  10  2  5  16  16  15 

84  11   1  4  15  H)  10 

40  9  1  0  00  00  00 

80  10  1  1  11  11  11 


XXXV. 

18*  54'  44" 

20  25   30 

87  30   10 

00  11 

27  29 


11 
34 


MOTION. 

XXXVI. 

26°  19'  15" 

19  26  20 

50  15  19 

33  10  11 

12  34  31 


xxxvir. 

10*.  24°  53'  .60" 
9   0  19   31 
3    9  23   42 
8   17  44   45 
7   10  10   10 


48  COMPOUND    SUBTRACTION. 

Q,  1.  What  does  Compound  Addition  teach? 

2.  How  do  you  place  the  different  denominations  iu 

Compound  Addition? 

3.  How  do  you  proceed  after  placing  the  denomina- 

tions under  each  other? 

4.  What  do  you  observe,  in  carrying  from  one  de- 

nomination to  another,  that  is  different  from 
Simple  Addition? 

5.  How  is  Compound  Addition  proved? 


COMPOUND  SUBTRACTION. 

Compound  Subtraction  teaches  to  find  the  difference 
between  any  two  sums  of  different  denominations. 
RULE. 

Place  those  numbers  under  each  other  which  are  of 
the  same  denomination — the  less  always  being  below 
the  greater.*  Begin  with  the  least  denomination,  and 
if  it  be  larger  than  the  figure  over  it,  consider  the  up- 
per one  as  having  as  many  added  to  it  as  make  one  of 
the  next  greater  denomination.  Subtract  the  lower 
from  the  upper  figure,  thus  increased,  and  set  down  the 
remainder,  remembering,  that  whenever  you  thus  make 
the  upper  figure  larger,  you  must  add  one  to  the  next 
superior  denomination. 

PROOF, 

As  in  Simple  Subtraction. 

EXAMPLES. 

ENGLISH    MONEY. 

I.  II  HI. 

£  s.  d.  qrs.  £  s.  d.  qrs.  £  s.  d.  qr. 
460  14  10  3  744  10  81  689  792 
320  10  8  2  398  18  10  3  372  18  4  3 


140       4       21       345     11        92       316       943 


*By  this  is  meant,  that  the  lower  line  of  figures  must  always  be  a  less  sum  than 
the  upper  line,  though  Borne  of  its  smaller  denominations  may  be  larger  than  those 
immediately  above  them,  in  the  upper  line. 


COMPOUND    SUBTRACTION.  49 

The  first  example  is,  in  itself,  sufficiently  plain.  In 
the  second,  finding  the  upper  figure  smaller  than  the 
lower  one,  as  it  is  in  farthings,  and  as  four  farthings  make 
a  penny,  1  suppose  four  added  to  the  upper  figure,  which 
makes  it  5.  Then  1  say,  3  from  5,  and  2  remain.  Placing 
the  2  underneath,  1  add  1  to  the  next  lower  figure,  name- 
ly, the  10,  which  thus  becom*  s  1 1 ;  and  as  the  8  standing 
above  is  less,  I  suppose  12  added  to  it,  which  makes  it 
20.  Taking  1 1  from  20,  9  remain.  Setting  the  9  un- 
derneath, and  adding  one  to  the  18,  it  becomes  19;  and 
as  the  upper  figure  is  smaller,  I  suppose  20  added  to  it, 
which  makes  it  30,  I  take  19  from  30,  and  1 1  remain. 
Placing  the  1 1  underneath,!  carry  one  to  the  next  figure, 
namely,  8;  and  then  proceed  as  in  Simple  Subtraction, 


TROY 

WEIGHT. 

IV 

V. 

Ib. 

oz. 

dwt. 

gr. 

Ib. 

oz. 

dwt. 

gr. 

947 

5 

13 

16 

876543 

7 

16 

11 

123 

10 

18 

20 

549876 

9 

17 

19 

AVOIRDUPOIS   WEIGHT. 

VI.  VII. 

"7 bra.  .crvt.    gr.      Ib.         Ton.    cwt.     qr.     Ib.  *     oz.      dr. 
5       13       1        12  8       16       0       24       11        11 

1       H       3       16  6       18       2       26       12       13 


APOTHECARIES   WEIGHT. 

i 

VIII.  IX. 

ft     3     3     6    gr.  ft       5     3     6    gr. 

44     7     5     1      12  87       4     1     0     10 

39     9     6     2     16  48     10     4     1      18 


COMPOUND    SUBTRACTION. 
LONG   MEASURE. 


X. 


deg.  mil.  fur.po.  ft.  in. 
85  53  7  16  10  iO 
60  57  0  27  14  11 


XI. 

deg.  mil.  fur.  pa 
95  10  3  12 
79  44  6  13 


CUBICK,   OR   SOLID   MEASURE. 

XII.  XIII.  XIV. 

Ton  ft.   in.    yd.  ft.   in.    Cord  ft.   in. 
18  17  1040    40  10   940    874  110  112S 
11  21  1485    32  16  1080    499  120  1699 


LAND,  OR  SQUARE  MEASURE. 

XV.  XVI.  XVII. 


acr.  roo.  per. 
987  2  23 
798  3  28 


acr.  roo.  per. 
8423  1  36 


acr.  roo.  per. 
9432  3  12 


4123  0  10     7324  2  24 


XVIII. 


CLOTH   MEASURE, 
XIX.  XX. 


XXI. 


yd.  qr.  nl.     E.  E.   qr.  nl.     E.  Fl.  qr.  nl.   E.  Fr.  qr.  hi. 


45 

29 


537 
409 


2 
3 


567 
389 


945 
739 


XXII. 

bush.  pk.  qt. 
74     1      1 
42     3     2 


DRY   MEASURE. 

XXIII. 

bush.  pk.  qt. 
230  0  0 
199  2  1 


XXIV. 

bush.  pk.  qt. 

56     1  1 

28     3  3 


COMPOUND    SUBTRACTION.  M 

WINE   MEASURE, 

XXV.  XXVI. 

Tun.  hhd.  gal.  qt.  pt.  Tun.  hhd.  gal.  qt.  pt. 

482     1      16     1      1  654     2     12     1     0 

297     3     22     3     1  276     3     40     2     1 


ALE   OR  BEER   MEASURE. 

XXVII.  XXVITI. 

hhd.  gal.  qt.  'pt.  hhd.   gal.  qt.  pt, 

8240  12  1   1  11917400  10  0  0 

1987  52  2  2  11654000  27  2  2 


TIME. 

XXIX.  XXX. 

w.  d.  h.   m.   s.       y.  mo.  w.  d.  h.  m.   $. 

8  2   12  42  30      20   10  1  4   10  27  37 

7   1   16  54  40      11   11  3  5  17  40  54 


MOTION. 

XXXI. 

16°  15'  35" 
12    45    48 

XXXII. 

8$.  10°  10'  10" 
6      15    50    30 

XXXIII. 

7s.     8°  37'  47" 
4      11     44    55 

Application  of  the  two  preceding  rules. 

1.  A  B  &  C  purchased  goods  in  partnership.     A  paid 
12  pounds,  10  shillings  and  8  pence;  B  paid  124  pounds* 
16  shillings;  and  C  paid  87  pounds  and   11  pence. — 
What  was  the  whole  amount  paid?      Ans.  £224  7s.  7d. 

2.  A  merchant  has  money  due  him: — from  one  man, 
587  pounds;  from  another,  420  pounds,  17  shillings  and 
6  pence;  from  a  third,  200  pounds;  and  from  a  fourth, 
978  pounds,  16  shillings  and  8  pence.     How  much  had 
he  due  in  all?  Ans,  £2186  14s,  2d. 


52  COMPOUND    MULTIPLICATION. 

3.  From  20  pounds,  take  12  pounds,  19  shillings  anct 
3  farthings.  Ans.  £7  Os.  lid.  Iqr 

4.  From  22  pounds,  take  19  shillings  and  1  farthing. 

Ans.  £21  Os.  lid.  3qrs. 

6.  From  17  pounds,  take  9  pounds,  9  shillings  and  9 
pence.  Ans.  £7  10s.  3d. 

6.  A  has  paid  B  £7  2s.  3d.,  £19   J  Is.  4d,  and  £17 
18s.  9^d.  on  account  of  a  debt  of  £60.     How  much  re- 
mains unpaid?  Ans.  £15  7s.  7  id. 

7.  A  ropemaker  received  3  tons,  4  cwt,  2  quarters, 
and  5  pounds  of  hemp;  of  which  he  made  into  cordage 
2  tons,  9  cwt,  and  1  quarter.     How  much  had  he  left? 

Ans.  15ewt.  Iqr.  5lbs. 

Q.  1.  What  does  Compound  Subtraction  teach? 

2.  How  do  you  set  down  Compound  Subtraction? 

3.  What  do  you  do  when  the  lower  denomination  is 

larger  than  the  one  that  is  above  it? 

4.  How  is  Compound  Subtraction  proved? 


COMPOUND  MULTIPLICATION. 

Compound  Multiplication  teaches  how  to  find  the  value 
of  any  given  number  of  different  denominations,  re- 
peated a  certain  number  of  times.  It  is  of  great  use  in 
finding  the  value  of  goods,  which  is  generally  done  by 
multiplying  the  price  by  the  quantity. 

CASE    I. 

When  the  quantity  or  multiplier  does  not  exceed  12. 
Set  down  the  price  of  1,  and  place  the  multiplier  un- 
der the  lowest  denomination;  and  in  multiplying  by  it, 
observe  the  same  rules  for  carrying  from  one  denomina- 
tion to  another  as  in  Compound  Addition. 

PROOF. 

Double  tbe  multiplicand,  and  multiply  by  half  the 
multiplier:  or,  divide  the  product  by  the  multiplier. 


COMPOUND   MULTIPLICATION*  53 

EXAMPLES. 
ENGLISH  MONEY. 

I.  What  will  7  yards  of  cloth  cost,  at  £1  12s.  lOfd 
per  yard?  7 


£11  10s.  3id, 

In  this  example,  I  say  7  time  3  make  21— -that  is,  21 
farthings,  equal  to  five  pence  and  one  farthing.  I  set 
down  the  one  farthing  under  the  place  of  farthings,  and 
carry  the  five  pence  to  the  place  of  pence  saying,  7  times 
10  arc  70,  and  5  make  75  pence — equal  to  6  shillings  and 
3  pence.  I  set  down  the  3  pence  under  the  pence  in  the 
sum  and  carry  the  6  shillings,  saying,  7  times  12  are  84, 
and  6  make  90  shillings,  equal  to  4  pounds  and  10  shillings. 
Setting  down  the  10  shillings  under  the  shillings,  I  carry 
the  4  pounds,  saying  7  times  1  are  7,  and  4  make  1 1 
pounds,  making  the  answer  to  the  question  1 1  pounds, 
10  shillings,  3  pence  and  1  farthing, 

II.  III.  IV. 

£    s.     d.  £    s.    d.  £    *,     A 

Multiply  4     14     lOf  7     12     9  14     15     9J 

by  2  4  8 


V.  VI.  VII. 

£     s.  d.  £     s.  d.  £      s.  d. 

Multiply  14     17  81  24     16  101  50     15  5| 

by  9  7  12 


TROY   WEIGHT. 

VIII.  IX. 

Ib.    oz.  dwt.  gr.  Ib.  oz.  dwt.  gr, 

Multiply  11     9     16     10  17     5     12     6 

by  4  5 


5* 


54  COMPOUND    MULTIPLICATION. 

AVOIRDUPOIS   WEIGHT. 


Mult.  3 

by 


x.  xi. 

Ton.  cwt.  qr.  Ib.  oz.    dr.  Ton.  cwt.  qr.  Ib.    oz.  dr. 

.  3      11     3     10     5     4  6     17     3  13     2      15 

fi  ft 


APOTHECARIES  WEIGHT. 

XII.  XIII. 

ft      3     3     6    gr.  fe     3     3     6    g-r. 

Mult.  43     10     6     2     10  4     10     7     2     13 

by                                  5  6 


LONG   MEASURE. 

XIV.  XV. 

cleg.  in.  fur.  p.    yd.  ft.    in.  L.  m.  fur.  p. 

Mult.  7     22     6     20     2     2     10  15     2     7     30 

by  26 


CUBICK,  OR   SOLID   MEASURE. 

XVI.  XVII.                           XVIII. 

Ton.  ft.     in.  yd.  ft.    in.       Cord  ft.     in. 

Mult.  10     16     15  14     2     19.        24     13     18 

by                      2  4                           6 


LAND,   OR   SQUARE   MEASURE. 

XIX.  XX.  XXI. 

A.    R.    P.          A.    R.    P.  A.    R.    P. 

Mult.  20     3     12  37     2     15  47     1      18 

by  2  4  6 


COMPOUND    MULTIPLICATION. 


55 


CLOTH    MEASURE. 

XXII.  XXIII.  XXIV.  XXV. 

yd.  qr.  nl  ELFr.gr.  nl  El.Fl.qr.nl.  El.E.  qr.nl 

Mult.  17     3     3       32     2     1       42     3     1        53     2     1 

by  4  6  8  9 


XXVI. 

bush.pk.  qt. 
Mult.  637 
by  5 


DRY    MEASURE. 

XXVII. 

bush.  pk.  qt. 

14     3     2 

6 


XXVIII. 

bush.  pk.  qt, 

34     2     3 

8 


WINE   MEASURE. 

XXIX.  XXX. 

Tun.  hhd.  gal.    qt.     pt.        Tun.  hhd.  gal.     qt.    pt. 

Mult.  1       2       12       3       1  2       3       40       3       1 

by  4  10 


ALE,   OR   BEER   MEASURE. 


XXXI. 


XXXII. 


hhd.  gal.     qt.    pt. 

Mult.  3       12       2       1 

by  5 


hhd.  gal.    qt.    pt. 
4       15       3        1 


XXXIII. 

y.  mo.  w.  d. 

Mult.  7735 

by  9 


XXXV. 

Mult.  24°     19'     11" 
by  10 


TIME. 

XXXIV. 

y<  mo.  w.  dt     h.     m.     s. 
8     5     3     6     20     32     10 


MOTION. 

XXXVI. 

Ws.     30°     17'     lOf" 
12 


tOMPOUND    MULTIPLICATION. 


CASE   II. 

When  the  multiplier  or  quantity  exceeds  12.  and  is  the  pro 
duct  of  two  factors  in  the  Multiplication  Table-  that  is, 
of  two  numbers  which  being  multiplied  together,  amount 
to  the  same  as  the  multiplier. 
Multiply  the  sum  by  one  of  the  two  numbers,  and  then 

multiply  the  product  by  the  other. 

EXAMPLES. 

I.  II. 

£    s.      d.  £     s.  d. 

Multiply  8     18     llfbylS.  13     12  9J  by  27, 

6  9 


53     13     10i 
3 

161        1       7J 

£    s.    d. 


122     15 


368 


£     *.  d. 


3.  Multiply  10  10  10    by  14.    Product  147  11   8 


4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 
12, 


£< 

a 


11  11  11  by  15. 

12  12  9  by  24. 
513  41  by  28. 
4  15  10"  by  42. 
7  17  7|  by  64. 
6  10  3  by  72. 
9  19  llf  by  81. 

10  15  9|  by  84. 
3  11  7£  by  96. 


173  18  9 
303  6  0 
158  14  6 
201  5  0 
504  9  4 
468  18  0 
809  16  7  A 
906  8  3 
343  16  0 


CASE  III. 

When  the  quantity,  of  multiplier,  is  such  a  number  that  no 
two  numbers  in  the  Multiplication  Table  will  produce  it. 

Multiply  the  sum  by  two  numbers  whose  product  will 
amount  to  nearly  the  same  as  the  multiplier;  then  mul- 
tiply the  sum  by  the  number  which  will  make  the  pro- 
duct of  the  two  numbers  equal  to  the  multiplier,  and 
add  its  product  to  the  sum  produced  by  the  two  num- 
bers. 


COMPOUND    MULTIPLICATION.  57 

EXAMPLES. 
I. 

£     s.     d.  £     s.       d. 

Multiply  7     10     5  7     10       5 

by  62  10  2 


75       4     2  15       0     10 

6 


451       50         Here  note,  I  multiply  by  10,  then 
15       0  10     by  6,  because  10  times  6  make  60; 

then  I  multiply  the  same  sum  by 

466       5  10     2,  that  [  multiplied,  first,  by  10, 

—    and  add  its  product  to  the  other 

product,  which  makes  the  amount  ®f  the  answer. 

£  *.    d.  £    s.    d. 

2.  Multiply  2   10  10    by  31.    Product  78  15   10 

3.  "  3   II  11     by  38.  "  136  12   10 

4.  "  411  21  by  68.  «  310     0     9 

5.  "  1      8  8    by  26.  «  37     5     4 

6.  "  13  3£  by  47.  «  54  14     81 

7.  «  12  41  by  83.  «  ^92  17     li 

CASE   IV. 

When  the  multiplier  is  greater  than  the  product  of  any  two 

numbers  in  the  Multiplication  Table. 
Multiply  the  sum  by  10,  and  that  product  by  10,  which 
is  equal  to  multiplying  by  100;  then  multiply  the  pro- 
duct by  the  number  of  hundreds  in  the  multiplier,  and  if 
the  sum  be  even  hundreds,  the  product  will  be  the  answer. 
If  there  be  odd  numbers  over  even  hundreds,  as  70, 80, or 
87,  &c.,  multiply  the  amount  or  product  of  the  first  mul- 
tiplication by  10,  by  the  number  of  tens  over  100;  thus, 
if  the»-e  be  70  over,  multiply  by  7.  If,  in  addition  to 
ten«.  there  are  smaller  numbers,  as  7,  8,  5,  &c,,  the  sum 
must  be  multiplied  by  such  number;  and  the  amount  of 
aU  the  multiplications  being  then  added  together,  their 
sum  will  be  the  answer, 


58  COMPOUND    MULTIPLICATION* 

EXAMPLES. 


Multiply 
by  4321 


£ 

s. 

d. 

i 

2 

4 

10 

11 

3 

4  amount  of 

10. 

10 

Ill       13         4  amount  of  100, 
10 


1116       13         4  amount  of  1000, 
4 


4466       13         4  amount  of  4000. 

335       00         0  amount  of  300. 

23         9         0  amount  of  21, 


4825         2         4  Answer. 


In  the  foregoing  example,  1  first  multiply  by  10,  three 
times,  which  gives  the  amount  of  the  sum  multiplied  by 
1000;  then  by  4,  which  gives  the  amount  of  4000.  The 
sum  is  yet  to  be  multiplied  by  32 1 .  To  do  this,  I  take  the 
product  of  the  sum  multiplied  by  100,  namely,  111/. 5s. 
and  multiply  it  by  3,  which  gives  the  product  of  the 
sum  by  300.  But  as  there  are  21  to  multiply  by,  1  take 
the  original  sum  and  multiply  it  by  7,  and  then  by  3; 
and  then  adding  the  products  together,  I  obtain  the  an- 
swer. 

£.    s.    d.  £     s.     d. 

-2.  Multiply          1     4     by     190.     Product       12  13     4 

3.  «          1     2     3     by     430.  "  478     7     6 

4.  «  7     6     by     506.  "  189   15     0 

5.  "  88     by     684.  «  296     8     0 

6.  «         1     3     9     by     375.          "          445     6     3 

7.  «  12     by  3456.  «          201    12     0 

APPLICATION. 

1.  What  do  B4  pounds  of  sugar  cost  at  9d.  per  pound? 


COMPOUND    MULTIPLICATION.  59 

2.  What  do  18  yards  of  cloth  cost  at  19s.  per  yard? 

3.  Sold  7  tons  of  iron  at  £32  10s.  per  ton:  how  much 
'is  the  amount?  Ans.  £227  10s.; 

4.  What  is  the  weight  of  4  hogsheads  of  sugar,  each 
weighing  7  cwt.  3qns.  191b?        Ans.  31cwt.  2qrs.  201bs. 

5.  What  is  the  weight  of  6  chests  of  tea,  each  weigh 
ing  3cwt.  2qrs.  91bs?  Ans.  2  Icwt.  Iqr.  261bs. 

6.  What  is  the  value  of  79  bushels  of  wheat,  at  11s. 
5fd.  per  bushel?  Ans.  £45  6s.  101. 

7.  What  is  the  value  of  94  barrels  of  cider,  at  12s. 
2d.  per  barrel?  Ans.  £57  3s.  8d. 

8.  What  is  the  value  of  114  yards  of  cloth  at  15s. 
3fd.  per  yard?  Ans.  £87  5s.  7Jd. 

9.  What  is  the  value  of  12  cwt.  of  sugar,  at  £3  7s. 
4d.  per  cwt.?  Ans.  £40  8s. 

10.  What  is  the  worth  of  63  gallons  of  oil  at  2s.  3d. 
per  gallon?  Ans.  £7  Is.  9d. 

11.  What  is  the  amount  of  120  days  wa§:es  at  5s.  9d. 
per  day?  Ans.  £34  10ss 

12.  What  is  the  worth  of  144  reams  of  paper  at  13s. 
4d.  per  ream?  Ans.  £96. 

J3.  What  will  Icwt.  of  sugar  cost,  at  lOfd.  per  lb.?* 

Ans.  £5  Os.  4d. 

14.  If  I  have  9  fields,  each  containing  12  acres,  2  roods 
and  25  poles;  how  many  acres  have  I  in  the  whole? 

Ans.  113A.  3R.  25P. 

Q.  1.  What  does  Compound  Multiplication  teach? 

2.  In  what  is  it  particularly  useful? 

3.  Which  is  made  the  multiplier — the  price  or  the 

quantity? 

4.  How  do  you  proceed  when  the  multiplier  does  not 

exceed  12? 

5.  How  Ho  you  proceed  when  the  multiplier  exceeds 

12? 

6.  When  the  multiplier  consists  of  no  two  component 

numbers,  as  in  case  third,  how  do  you  proceed? 

7.  How  do  you  proceed  in  case  fourth? 

8.  How  is  Compound  Multiplication  proved? 

*ft  must  be  recoHected  that  Icwt.  is  1121bs. 


£0  COMPOUND    DIVISION. 

COMPOUND  DIVISION. 

Compound  Division  teaches  the  manner  ©f  dividing 
numbers  of  different  denominations. 

CASE    I. 

When  the  divisor  does  not  exceed  12. 

Begin  at  the  highest  denomination,  and  after  dividing 
that,  if  any  thing  remain,  reduce  it  to  the  next  lower 
denomination,  adding  it  to  that  denomination  in  the  sum, 
and  proceed  in  this  manner  until  the  whole  is  divided. 
If  the  number  of  either  denomination  should  be  too 
small  to  contain  the  divisor,  reduce  it  to  the  next  lower 
denomination,  and  add  it  thereto,  asxlirected  in  case  of 
a  remainder.  The  denominations  in  the  quotient  must 
be  kept  separate. 

PROOF. 

Multiply  the  quotient  by  the  divisor,  and  the  product, 
if  right,  will  be  equal  to  the  dividend. 

EXAMPLES. 
I.  II.  III. 

£    s.    d.  £    s.     d.  £    s.     d. 

Divide  2)6     88         4)3     3     10         5)7     2     3 


344  15     111  is     51+3 

IV.  V.  VI. 

£    s.     d.  £  s.  d.  £  s.      d. 

5)6     17     11         6)9  9  9         12)21  16     Hi 


177  1    11     7i  1      16       4f-flO 


In  doing  the  6th  sum,  which  is  divided  by  12.  I  find 
the  divisor  is  contained  once  in  21;  and  setting  down  1 
1  find  9  pounds  remaining;  which,  reduced  to  shillings, 
and  added  to  the  16  shillings  in  the  sum,  make  196 
Shillings.  The  divisor  being  contained  16  times  in  196, 
with  4  remaining,  I  set  down  16,  and  reducing  the  4 
shillings  to  pence,  and  adding  them  to  the  1 1  pence,  in 
the  sum,  the  amount  is  59  pence.  The  divisor  is  con- 


COMPOUND   DIVISION.  61 

lamed  4  times  in  59,  leaving  1 1  pence  remaining.  I  sef 
down  4,  and  the  remaining  1 1  pence  reduced  to  farthings 
and  added  to  the  half  penny  or  2  farthings  in  the  sum, 
make  46  farthings;  and  as  the  divisor  is  contained  3 
times  in  46,  leaving  a  remainder  of  10, 1  set  down  £  and 
place  the  final  remainder  at  the  right  hand  of  the  sum. 


£ 

s. 

d. 

£  *. 

d. 

7. 

Divide 

12 

10 

10 

by 

5. 

Quotient 

2 

10 

2 

8. 

u 

13 

13 

9 

by 

4. 

" 

3 

8 

5J 

9. 

u 

2 

18 

H£ 

by 

3. 

u 

19 

7f 

10. 

u 

7 

7 

7 

by 

4, 

u 

1 

16 

10  J 

11. 

" 

177 

19 

llf 

by 

12. 

«       14 

16 

7f 

CASE    II. 

ie  divisor  exceeds  12,  cmrf  w  tfta  product  of  two 

numbers  multiplied  together. 
Divide  by  one  of  the  numbers:  then  divide  the  quo- 
tient by  the  other. 

EXAMPLES. 

Divide  £5  10s.  6d.  by  48, 

£  \    d. 

6)5     10     6 

8)    18     5 

2     3+5  Answer. 

Note. — If  there  be  any  remainder  in  the  first  opera- 
tion, and  not  any  in  the  second,  it  is  the  true  one.  When 
there  is  a  remainder  in  the  second  operation,  multiply  it 
by  the  first  divisor,  and  add  it  to  the  first  remainder,  if 
there  be  any,  and  it  forms  the  true  remainder. 

£     s.     d.  £    s.    d. 

2.  Divide  240  12  10    by  16.  Quotient  15     0     9^+4 

3.  «         88  13   11     by  21.        «  44     5|+2 

4.  «         90  15     41  by  32.        «  2   16     8f 

5.  «       450     8     8    by  42.        "          10  14     5|+26 

6.  ,  «     .789   19     9    by  64.        "     •     12     6 

7.  «       840     4    3£by  72.        «         11   13 


62  COMPOUND   DIVISION, 

CASE   III. 

When  the  divisor  is  more  than  12,  and  can  not  be  produced 

by  multiplying  any  two  numbers  together. 
Divide  after  the  manner  of  Long  Division,  reducing 
from  higher  to  lower  denominations,  as  in  the  following 

EXAMPLES, 

Divide  £61  12s.  by  23. 
i. 

£     *• 

23)61  12(£2  13s.  6d.  3qrs.-f-3  Ans. 
46 

—  Divide  £14  10s.  llfd.  by  95. 

15  ii. 

20  X  £      s.      d. 

95)14     10     11£(£0  3s.  0£d.+2   Ans, 

312  20X 

28  

-*-  290 

82  285 

69  

—  5 

13  12X 

12X 

71 

156  4-X 
138 

287 

18  285 

4X 

2 

72 
69 

3 

Note. — In  the  second  example,  1  tind  the  divisor  grp a- 
ter  than  the  number  of  pounds  in  the  dividend.  I  there- 
fore set  down  a  cypher  in  the  place  of  pounds  in  the 
quotient,  then  reduce  the  14  pounds  in  the  sum,  inte 


COMPOUND   DIVISION.  83 

(•.hillings,  at  the  same  time  adding  the  10  shillings  in  the 
sum,  to  the  amount,  which  therehy  becomes  290.  In 
290  the  divisor  is  contained  3  times,  and  5  over.  This 
6  shillings  1  multiply  by  12,  to  reduce  it  to  pence,  adding 
to  it  the  1 1  pence  in  the  sum;  and  the  amount  being  still 
smaller  than  the. divisor,  1  set  down  a  cypher  in  the 
place  of  pence,  in  the  quotient,  and  reduce  it  to  far- 
things ;% which,  with  the  3  farthings  in  the  sum,  amounts 
to  287  farthings.  In  287  the  divisor  is  found  three 
times,  and  there  is  a  remainder  of  2.  The  quotient, 
therefore,  contains  a  cypher  in  the  place  of  pounds;  3, 
in  the  place  of  shillings;  a  cypher  in  the  place  of  pence, 
and  3  in  the  place  of  farthings. 

Though  this  operation  is  longer,  it  is,  perhaps,  less 
liable  to  error  than  either  of  the  preceding  cases. 

£   ,.   a.  £.   s.   d. 

3.  Divide  20  10     8    by  17.  Quotient  1     4     l£+9 

4.  "  27   18  7    by  29.  "  0  19  3-»+6 

5.  «  147     4  4    by  65.  «  25  31+18 

6.  "  581   19  ll£by  73.  «  7   19  51+49 

7.  "  77     3  3f  by   19.  «  41  21+17 

8.  «  319     7  101  by  29.  «  11     0  3i+l 

APPLICATION. 

1.  If  42  cows  cost  £126  16s.  6d;  what  was  the  price 
of  each?  Ans.  £3  Os.  4id, 

2.  If  £1000  be  divided,  equally,  among  40  men;— 
what  will  each  receive?  Ans.  £25* 

3.  Five  men  bought  a  quantity  of  hay,  weighing  21 
tons.  13  hundred,  and  3  quarters;  which  they  divided, 
equally  among  them.     What  was  the  share  of  each£ 

Ans  4  tons,  6cwt.  3qrs. 

4.  A  farmer  had  3  sons,  to  whom  he  gave  a  tract  of 
land  containing  520  acres,  3  roods,  29  perches;  and  the 
land  was  too  be  divided,,  equally  among  them.     What 
was  the  portion  of  each?  Ans.  173A.  2R.  23P: 

5.  Divide  375  miles,  2  furlongs,  7  poles,  2  yards,  1 
foot,  2  inches,  by  39.      Ans.  9M.  4fur.  39P.  Oyd.  2ft.  8m. 

6.  Divide  571  yards,  2  quarters,  1  nail  by  47. 

Ans,  12yds.  Oqr.  2na, 


64  REDUCTION. 

7.  Divide  120  months,  2  weeks,  3  days,  5  hours,  20 
minutes,  by  111.  Ans.  Imo.  OW.  2D.  10H.  12  min, 

8.  Divide  54  dollars,  54  cents,  4  mills,  among  3  girls 
and  2  boys;  and  give  to  each  girl  twice  as  much  as  to 
each  boy.     What  does  each  girl  receive? 

Ans.  $13  63c.  6m, 

9.  Divide  $20  among  8  persons;  and  give  the  first 
JOc.  more  than  the  second;  the  second  lOc.  more  than 
the  third,  Sue. ;  what  sum  does  the  eighth  person  receive? 

Ans.  $2.15. 

Q.  1.  What  does  Compound  Division  teach? 

2.  How  do  you  proceed  when  the  divisor  does  not 

exceed  12? 

3.  What  do  you  do  when  the  number  of  either  de-- 

nomination is  too  small  to  contain  the  divisor? 

4.  What  do  you  eto  when  the  divisor  exceeds  12,  and 

is  the  product  of  two  numbers  multiplied  to- 
gether? 

5.  How  do  you  proceed  when  the  divisor  is  more 

than  12,  and  can  not  be  produced  by  multih 
plying  any  two  numbers  together? 
$.  How  is  Compound  Division  proved? 


REDUCTION. 

Reduction  teaches  to  change  numbers  of  one  denomfc 
nation  into  those  of  other  denominations,  retaining  the 
same  value.  Its  operations  are  performed  by  Multipli- 
cation and  Division.  When  performed  by  Multiplica- 
tion, it  is  called  Reduction  Descending,  when  performed 
by  Division,  it  is  called  Reduction  Ascending. 

1 .  How  many  farthings  will  it  take  to  make  two  pence  ? 
How  many  pence  to  make  two  shillings? — How  many 
shillings  to  make  two  pounds? 

2.  [low  many  gills  to  make  three  pints? — How  many 
pints  to  make  three  quarts? — How  many  quarts  to  make 
three  gallons? 

3.  How  many  quarts  to  make  four  pecks? — How  ma- 
ny pecks  to  make  four  bushels? 


REDUCTION.  65 

4.  How  many  pence  are  there  in  eight  farthings? — 
How  many  shillings  in  twenty- four  pence? — How  many 
pounds  in  forty  shillings? 

5.  How  many  pints  in  twelve  gills? — How  many  quarts 
in  six  pints? — How  many  gallons  in  twelve  quarts? 

6.  How  many  pecks  in  thirty-two  quarts? — How  ma- 
ny bushels  in  sixteen  pecks? 

7.  How  many  pounds  and  shillings  in  thirty  shillings? 
How  many  shillings  and  pence  in  thirty  pence? 


DEDUCTION   DESCENDING, 

RULE. 

Multiply  the  numbers  in  the  highest  denomination 
given,  by  the  number  that  it  takes  of  the  next  less  de- 
nomination to  make,  one  of  that  greater;  and  thus  pro- 
ceed until  you  shall  have  multiplied  each  higher  deno- 
mination by  the  number  that  it  takes  to  form  the  next 
lower,  until  you  come  to  the  lowest  of  all. 
PROOF. 

Descending  and  Ascending  Reduction   prove  each 
other. 

SIMPLE    EXAMPLES. 

r. 

Reduce  25  pounds  to  shillings.         Ans.  500  shillings* 
25 
20  shillings  in  a  pound. 

500  shillings, 


Reduce  50  shillings  to  pence,  Ans  £00  pence 

50 
12  pence  in  a  shilling. 

600  pence. 
6* 


66  REDUCTION. 

III. 

Reduce  15  pence  to  farthings.          Ans.  60  farthings 
1  o 
4  farthings  in  a  penny, 

60  farthings. 

IV. 

Reduce  10  tons  to  hundred  weights,        Ans.  200cwt- 
10 
20  hundred  in  a  ton* 

*  200  hundred. 

v. 

Reduce  36  peunds  to  ounces.  Ans.  576  ounces; 

36 
16  ounces  in  a  pound, 

216 
36 

576  ounces. 

6.  Reduce  70  miles  to  furlongs,  Ans.  560  fur, 

7.  Bring  30  furlongs  to  rods.  Ans.  1200  rods. 

8.  Bring  20  rods  to  feet.  Ans.  330  feet 

9.  Bring  24  feet  to  inches.  Ans.  288  inches. 

10.  Reduce  32  acres  to  roods.  Ans.  128  roods. 

11.  Bring  24  square  perches  to  square  yards. 

Ans.  726  square  yards. 

12.  Reduce  10  hogsheads  to  gallons.          Ans.  630gal. 

13.  Bring  25  gallons  to  pints.  Ans.  200  pints. 

14.  Reduce  23  bushels  to  pecks.  Ans.  92  pecks. 

15.  Bring  12  pecks  to  pints.  Ans.  192  pints. 

16.  Reduce  15  years  to  months.          Ans.  180  months. 

17.  Bring  75  days  to  hours.  Ans.  1800  hours. 

18.  Bring  24  hours  to  minutes.          Ans.  1440  minutes, 

19.  Bring  10  signs  to  degrees.  Ans.  600  degrees 


REDUCTION.  67 

COMPOUND  EXAMPLES, 

I. 

£     s.     d.    qfs. 

In  15         17         11         3  how  many  farthings,? 
20  shillings  in  a  pound. 

317  shillings. 
12  pence  in  a  shilling. 


3815  pence. 

4  farthings  in  a  penny. 


15263  farthings. 

Note. — In  multiplying  by  20,  I  added  in  the  17  shil- 
lings, by  12,  the  11  pence;  and  by  4,  the  3  farthings^ 
and  this  must  be  observed  in  all  similar  cases. 

To  prove  this  sum,  let  the  order  of  it  be  changed, 
and  it  will  stand  thus:  in  15263  farthings,  how  many 
pounds? 

4)15263 

12)3815+3  quarters. 
210)31)7+11  pence. 

£15  17s.  lid.  3qrs.  Ans. 

In  reducing  Federal  Money  from  a  higher  to  a  lower 
denomination,  it  is  only  necessary  to  annex  as  many  cy- 
phers as  there  are  places  from  the  denomination  given 
to  that  required;  or,  if  the  given  sum  be  of  different 
denominations,  annex  the  figures  of  the  several  denomi- 
nations in  their  order,  and  continue  with  cyphers,  whep 
the  sum  requires  it,  to  the  denomination  intended. 

2.  Thus,  in  7  eagles,  3  dollars,  how  many  mills? 

Ans.  73000. 

3.  In  85  dollars,  how  many  mills?  Ans.  85000. 

4.  In  574  eagles,  how  many  dollars?  Ans.  5740. 

5.  In  469  dollars,  bow  many  cents?  Ans.  48900, 


68  REDUCTION. 

6.  In  844  dollars,  75  cents,  how  many  mills? 

Ans.  844750 

7.  In  10QO  dollars,  how  many  mills?      Ans.  1000000, 

8.  In  25  dollars,  47  cents,  8  mills;  how  many  mills? 

9.  In  29  guineas  at  28s.  each,  how  many  pence t' 

Ans.  97*4. 

10.  In  20  acres,  29  poles,  or  perches,  how  many  square 

perches?  Ans.  3229, 

11.  How  many  solid  feet  in  30  cords  of  wood? 

Ans.  3840. 

12.  How  many  grains  in  100  Ibs.— Troy  Weight? 

Ans.  576000. 

13.  How  many  Ibs.  in  a  ton :— -Avoirdupois  Weight? 

Ans.  2240. 

14.  In  27  Ibs. — Apothecaries  Weight;  how  many  grains? 

Ans.  155520. 

15.  In  30  yards,  how  many  nails?  Ans.  480. 

16.  In   360   degrees,  being   the   distance  round   the 
world,  how  many  inches,  allowing  69J  miles  to  a  de- 
gree? Ans.  1,587,267,200. 

17.  How  many  pints  are  there  in  one  tun  of  wine? 

Ans.  2016. 

18.  How  many  half  pints  in  one  hogshead  of  beer? 

19.  How  many  pints  in  400  bushels?  Ans.  25600. 

20.  How  many  seconds  in  80  years? 

Ans.  2,524,554,960; 

21.  How  many  yards  in  4567  miles?        Ans.  8037920. 

22.  In  £20  17s.,  how  many  pence  and  half  pence? 

Ans.  5004  pence,  and  10,008  half  pence, 


REDUCTION   ASCENDING. 

RULE. 

Divide  the  figure  or  figures  in  the  lowest  denomina 
tion,  by  so  many  of  that  name  as  make  one  of  the  next 
higher;  and  continue  the  division  until  you  have  brought 
it  ioto  that  denomination  which  your  question  requires. 

In  reducing  Federal  Money  from  a  lower  to  a  higher 
denomination,  nothing  more  is  necessary  than  to  cut  off 


REDUCTION.  69 

so  many  places  on  the  right  hand  side  of  the  sum,  as 
there  are  denominations  lower  than  the  one  required, 
Thus,  98765  mills  are  reduced  to  dollars,  cents,  and 
mills,  by  cutting  off  one.  figure  for  mills,  two  more  for 
cents,  and  the  remaining  figures  being  dollars,  the 
amount  is  $98|76|5 — or  ninety-eight  dollars,  seventy-six 
cents,  five  mills. 

SIMPLE    EXAMPLES. 

1.  How  many  dollars  are  there  in  8000  mills? 

8|00|0  Ans.' 8, 

2.  In  487525  cents,  hpw  many  dollars  and  cents? 

4875125  Ans.  $4875.25. 

3.  In  999888  mills,  haw  many  dollars,  cents,  and  mills  ? 

999J88J8  Ans.  jj999.8a.87 

4.  In  19200  farthings,  how  many  pounds? 

4)19200 

12)4800 


20)400 

Ans.  20  p 

5.  In  480  nails,  how  many  yards? 

4)480 

4)120 
30  Ans. 

COMPOUND    EXAMPLES. 

6.  la  52300  farthings,  how  many  pounds? 

4)52300 

12)13075 
2|0)108|9+7 


Ans.  £54  9s.  7d, 

7.  In  8428  Ibs.  Avoirdupois  Weight,  how  many  tons? 
Ans.  3  tons.  18cwt.  3qrs.  81bs, 


.  0  REDUCTION. 

8.  In  524  Ibs.  Avoirdupois  Weight,  how  many  cwt.  &,c, 

Ans.  4cwt.  2qrs.  SOlbe, 

9.  In  125440  grains,  Troy  Weight,  how  many  Ibs? 

Ans.  44, 

10.  In  155520  grains,  Apothecaries  Weight,  how  many 
pounds?  Ans.  27. 

1 1 .  How  many  miles  are  there  in  1,585,267,200  inches? 

Ans.  25020. 

12.  In  4000  nails,  how  many  yards?  Ans.  250, 

13.  In  8000  square  rods,  how  many  acres?       Ans.  50. 

14.  In  2016  pints  of  wine,  how  many  tuns?.         Ans.  1, 

15.  How  many  bushels  are  there  in  80,000  quarts? 

Ans.  2500. 

16.  In  2,524,554,960  seconds, how  many  years?  Ans.  80. 

17.  In  3840  solid  feet,  how  many  cords?  Ans.  30. 

18.  [n  1728  half  pints  of  beer,  how  many  hogsheads? 

Ans.  2. 

19.  Bring  240,000  pence  to  pounds.  Ans.  £1000, 

20.  Bring  112  quarters  to  cwt.  Ans.  28  cwt. 

21.  Bring  120  miles  into  leagues.  Ans.  40L, 

22.  Bring  1280  poles  into  furlongs.  Ans.  32  fur. 

23.  Reduce  960  nails  to  quarters.  Ans.  240  qrs. 

24.  Reduce  17280  cubick,  or  solid  inches,  to  solid  feet, 

Ans.  10  solid  feet. 

25.  In  768  pints,  how  many  bushels?  Ans.  12. 

26.  In  1890  gallons,  how  many  hogsheads?       Ans.  30, 

Q,  1.  What  does  Reduction  teach? 

2.  By  what  rules  are  its  operation  performed? 

3.  When  performed   by  multiplication,  what  is  it 

called? 

4.  What  is  your  rule  for  Reduction  Descending? 

5.  When  performed  by  Division  what  is  it  called? 

6.  What  is  your  rule  for  Reduction  Ascending? 

7.  How  do  you  reduce  Federal  Money  from  a  lower 

to  a  higher  denomination? 
3.  How  is  Reduction  proved  ? 


EXCHANGE.  71 

EXCHANGE. 

-  Exchange  teaches  to  change  a  sum  of  one  kind  of 
money  to  a  given  denomination  of  another  kind. 
T*o  reduce  the  currency  of  each  of  the  United  States  to  dol- 
lars and  cents,  or  Federal  Money. 

RULE. 

Reduce  the  sum  to  pence ;  to  the  pence  annex  two 
cyphers;  then  divide  by  the  number  of  pence  in  a  dol- 
lar, as  it  passes  in  each  State,  the  quotient  or  answer 
will  be  in  cents,  which  may  be  easily  reduced  to  dollars* 
Note. — This  rule  applies  to  the  currency  of  any  other 
country,  if  its  currency  be  in  pounds,  shillings,  pence,  &c. 
EXAMPLES. 

1.  Reduce  621  pounds,  New  England,  Virginia,  and 
Kentucky  currency,  to  dollars  and  cents  j  a  dollar  being 
72  pence, 

£ 

621 
20 

12420 
12 

72)14904000(^2070.00 
144 

504 
504 

000 

2.  Reduce  12  pounds,  3  shillings,  and  9  pence  to  dol- 
dollars  and  cents.  Ans.  $40.621, 

3.  Reduce  30  pounds  and  3  shillings  to  dollars  and 
cents.  Ans.  $100.50 

4.  In  £763  New  England  and  New  York  currencies, 
how  many  dollars,  cents,  and  mills? 

Ans.  $2543.33cts.  3m.  N.  E.  cur. 
$1907,50cts.  N,  Y.  cur. 


72  EXCHANGE. 

5.  In  9  pounds  and  16  shillings  in  New  York  and 
North  Carolina  currency,  how  many  dollars  and  cents, 
reckoning  96  pence  to  a  dollar? 

£       s. 

9         16 
20 


196 
12 

3X12=96  8)235200 

12)29400 

$24.50  Answer, 

6.  In  30  pounds,  how  many  dollars  and  cents? 

Ans.  $75.00. 

7.  In  27  pounds,  2  shilling,  how  many  doilars  and 
cents?  "  Ans.  $67.75. 

8.  fn  942  pounds  of  New  Jersey.  Pennsylvania,  Dela- 
ware, and  Maryland  currency ;  how  many  dollars  and 
Qents,.a  dollar  being  90  pence? 

•*"  942 

20 

18840 
12 

9|0)2260800jO 

$2512.00  Answer, 
"0.  In  12  pounds  how  many  dollars  and  cents? 

Ans.  $32. 

10.  In  £86  6s.  5£d.  how  many  dollars,  cents  and  mills? 

Ans.  $230.19cts.  4m. 

11.  In  21  pounds,  South  Carolina  and  Georgia,  curren- 
cy, how  many  dollars  and  cents,  there  being  56  pence 
in  a  dollar? 


EXCHANGE 

£ 

21 

20 

420 
12 

7X8=56)  7)504000 

8)720.00 

$90.00  Answer. 

12.  In  56  pounds,  how  many  dollars,  &c.?     Ans.  $240. 

13.  In  108  pounds,  Canada  and  Nova  Scotia  currency, 
iiow  many  dollars,  &c.,  there  being  60  pence  in  a  dollar? 

£ 

108 
20 

2160 
12 

6|0)25920.0|0 

$432.00  Answer. 

14.  In  460  pounds  and  16  shillings  sterling,  or  EnglisL 
money,  how  many  dollars,  &c.3  there  being  54  pence  ip 
a  dollar? 

£ 

460         16 
20 

9216 
12 


9X6=54         9)110592.00 
6)12288.00 


$2048.00  Answer, 
7 


74  EXCHANGE. 

15.  Reduce  16  pounds,  6  shillings,  and  3  pence,  English 
money,  to  dollars  and  cents. 

£        9.        d. 
16         6         3 
20 

326 
12 

9)3915.00 
6)435.00 

$72.50  Answer. 

To  bring  Federal  Money  into  pounds,  shillings  and  pence,^ 
RULE. 

Multiply  the  dollars,  or  dollars  and  cents,  by  the  num- 
ber of  pence  in  a  dollar  of  the  currency  to  which  you 
wish  to  change  the  given  sum; — the  answer  will  be  in 
pence,  which  can  then  be  reduced  to  shillings  and  pounds. 
When  there  are  cents  in  the  sum  to  be  reduced,  two  fi- 
gures must  be  cut  off  from  the  right  of  the  product,  be- 
fore bringing  it  into  pounds. 

Note. — This  rule  applies  to  the  currency  of  any  coun- 
try whose  currency  is  in  pounds,  shillings,  &c. 

EXAMPLES. 

1.  In  $16.50  how  many  pounds  and  shillings,  in  sterr 
ling,  or  English  Money,  a  dollar  being  four  shillings  and 
six  pence,  or  54  pence? 

$16.50 

9X6=54  9 


12)891.00 
2|0)7|4.-}-3 
£3  14s,  3d. 


EXCHANGE.  "75 


--    2.  In  33  dollars,  how  many  pounds,  &c.'? 
$33 
9 

297 
6 

12)1782 


2|0)14|8— 6d. 

£7.  8s.  6d,  Answer. 

3.  In  U)00000  dollars,  how  many  pounds  sterling? 
$  1000000 
9 

9000000 
6" 


12)54000000 
2|0)450000|0 

£225.000  Answer,* 

4.  Reduce  432  dollars  into  the  currency  of  Canada 
and  Nova  Scotia,  a  dollar  being  equal  to  five  shillings, 
or  60  pence. 

$432 
60 

12)25920 
3|0)216|0 

£108  Answejr. 

5.  In  $490,50  how  many  pounds,  shillings,  &c.? 

Ans.  £122  12s.  6d^ 

*  Federal  Money  may,  also,  be  changed  into  English  Money,  by  multiplying  the 
dollars  by  9,  and  dividing  the  product  by  40. 


T6>  VULGAR    FRACTIONS* 

6.  Bring  $150.25,  into  the  currency  of  New  England, 
Virginia,  and  Kentucky,  a  dollar,  being  equal  to  7£ 
pence. 

$150.25 
9 


9X8=72  135225 

8 

12)10818|00 
2|0)90l  I— 6 

£45.  Is.  6d.  Answer. 

Q-.  1.  What  does  Exchange  teach? 

2.  How  do  you  reduce  the  currency  of  any  one  of 

the  United  States  to  Federal  Money? 

3.  Does  this  rule  apply  to  the  currency  of  any  other 

country  ? 

4.  How  do  you  change  Federal  Money  into  pounds, 

shillings,  and  pence  of  any  state,  or  country? 
6.  Among  the  various  kinds  of  money,  what  kind  is 
the  most  easily  reckoned  £ 


VULGAR  FRACTIONS. 

^-^  Fractions  are  broken  numbers,  expressing  any  assigna- 
ble part  of  an  unit,  or  whole  number.  They  are  repre- 
sented by  two  numbers  placed  one  above  another,  with 
aline  drawn  between  them;  thus,  f,  f,  &c.  signifying 
4wo  fifths  and  five  eights. 

The  figure  above  the  line  is  called  the  numerator, 
and  that  below  the  line,  the  denominator.  The  denomi- 
nator shows  into  how  many  equ.il  parts  the  whole  quan- 
tity is  divided,  and  represents  the  divisor  in  division. — 
The  numerator  shows  how  many  of  those  parts  are  ex- 
pressed by  the  fraction;  being  the  rfr»,  .i^c'er  after  di- 
vision. Both  these  numbers  are  sometimes  calied  the 
terms  of  the  fraction. 


VULGAR    FRACTIONS.  !fl 

Questions  to  prepare  the  learner  for  this  rule. 

1.  If  a  pear  be  cut  into  two  equal  parts,  what  is  one 
of  those  parts  called?  Ans.  a  half. 

2.  If  you  cut  a  pear  into  three  equal  parts,  what  is- 
one  of  these  parts  called?  Ans.  one  third, 

3.  How  many  thirds  of  any  thing  make  the  whole? 

4.  If  a  pear  be  cut  into  four  equal  parts,  what  is  one 
of  those  parts  called?     Ans.  one  fourth.     What  are  two 
of  the  parts  called  ?     Ans.  two  fourths.     What  are  three 
of  them  called?  Ans.  three  fourths. 

5.  How  many  fourths  of  a  thing  make  the  whole? 

6.  If  an  orange  be  cut  into  rive  equal  parts,  what  i's 
one  of  the  parts  called?     Ans.  one  fifth.     What  are  two 
of  the  parts  called?     Ans.  two  fifths.     What  are  three 
of  them  called?     Ans.  three  fifths.     What  are  four  of 
them  called?  Ans.  four  fifths. 

7.  How  many  fifths  of  a  thing  make  the  whole? 

8.  If  you  cut  a  pear  into  six  equal  parts,  what  is  one 
of  the  parts  called?     What  are  two  of  them  called?- — 
What  are  three  of  them  called  ?     What  are  four  of  them 
called? 

9.  How  many  thirds  are  there  in  one?     How  many 
fourths?     If  four  fourths  make  the  whole,  what  part  are 
two  fourths?     What  part  of  three  is  one?     What  part 
of  four  are  two?     What  part  of  six  are  two?     What 
part  of  eight  are  two?     What  part  of  eight  are  six? — 
What  part  of  9  are  6?     What  part  of  10  are  2?     What 
part  of  10  are  4?     What  part  of  10  are  7?     What  part 
of  12  are  6?     What  part  of  12  are  4?     What  part  of 
12  are  3?     What  part  of  12  are  2? 

10.  How  many  are  two  fourths  of  12?  How  many  are 
three  fourths  of  12?  Two  thirds  of  12,  are  how  many? 
How  many  are  5  times  8?  In  one  eighth  of  40,  tow 
many  ?  In  three  eights  of  40,  how  many  ?  Four  eights 
of  40,  are  how  many  ?  Then  £  of  any  number,  or  of 
any  thing  amount  to  how  many,  or  how  much?  How 
many  are  f  of  30?  How  many  are  f  of  30?  How  ma- 
ny in  £  of  60?  In  £  of  60,  how  many?  In  J  of  60, 


78  VULGAR    FRACTIONS. 

how  many?  In  ^  ->f  60,  how  many?  How  many  are 
^  of  60?  Hew  many  are  £  of  60?  How  many  are  | 
of  60?  In  2  and  },  how  many  fifths?  In  5  and  |  how 
rtiany  fifths?  In  1  of  100  how  many?  In  £  of  100 
cents,  or  1  dollar,  how  much?  How  much  are  f  and  i? 
How  much  are  J  and  I?  How  much  are  f  and  -J?  How 
much  are  f  ?  How  much  are  ^  and  |?  How  much  are 
f,  J,  and  i?  In  ^  how  many?  In  f,  how  many?  If 
you  take  £  from  one  dollar,  how  much  will  remain?  If 
you  take  J  from  one  dollar,  how  much  will  remain?  If 
you  take  J  from  a  pound,  how  much  will  remain?  If 
you  take  £  from  one,  kow  much  will  remain?  How 
many  fourths  are  2  times  £?  How  many  are  5  times  f  ? 
How  many  are  3  times  f  ?  In  \-,  how  many?  In  J^, 
how  many? 

11.  If  one  half,  three  fourths,  and  a  quarter,  be  added, 
how  much  will  be  their  amount? 

12.  If  you  take  two  eights  from  eleven  eights,  how 
much  will  remain? 

13.  What  is  a  proper  fraction? 

Ans.  When  the  numerator  is  less  than  the  deaomina^ 
tpr,  as  i,  or  | ,  &c. 

14.  What  is  an  improper  fraction? 

Ans.  It  is  that  in  which  the  numerator  is  equal,  or 
superior  to  the  denominator;  as  f ,  or  f ,  or  J,  &c. 
45.  What  is  a  simple  fraction? 
Ans.  It  is  a  fraction  expressed  in  a  simple  form;  as, 

*»  4,  f  - 

16.  What  is  a  compound  fraction? 

Ans.  It  is  the  fraction  of  a  fraction,  or  several  frac- 
tions connected  together  with  the  word  of  between 
them;  as  |,  of  |,  of  |,  or  f  of  ^  &G.,  which  are  read 
thus,  one  half  of  two  thirds.  &e. 

17.  What  is  a  mixed  number? 

Ans.  It  is  composed  of  a  whole  number  and  a  frac- 
tion; as  3J.  or  12£. 

1 8.  What  is  the  commen  measure  of  two  or  more  num- 
bers? 

Ans.  It  is  that  number  which  will  divide  each  of  them 
without  a  remainder;  thus,  5  is  the  common  measure  of 


DEDUCTION  OF  VULGAR  FRACTIONS.       79 

10.  20,  and  30,  and  the  greatest  number  that  will  do  this, 
is  called  the  greatest  common  measure. 

19.  What  is  meant  by  the  common  multiple? 

Ans.  Any  number  which  can  be  measured  by  two  or 
more  numbers,  is  called  the  common  multiple  of  those 
numbers;  and  if  it  be  the  least  number  that  can  be  so 
measured,  it  is  called  the  least  common  multiple;  thus, 
40,  60,  80,  100,  are  multiples  of  4  and  5;  but  their 
least  common  multiple  is  20. 

20.  When  is  a  fraction  said  to  be  in  its  lowest  terms? 
Ans.  When  it  is  expressed  by  the  smallest  numbers 

possible. 

21.  What  is  meant  by  a  prime  number? 

Ans.  It  is  a  number  which  can  only  be  measured  by 
itself,  or,  an  unit. 

22.  What  is  meant  by  a  composite  number? 

Ans.  That  number,  which  is  produced  by  multiplyingr 
several  numbers  together,  is  called  a  composite  number. 

23.  What  is  a  perfect  number? 

Ans.  A  perfect  number  is  one  that  is  equal  to  the  sum 
of  its  aliquot  parts.* 


REDUCTION  OF  VULGAR  FRACTIONS. 

Reduction  of  Vulgar  Fractions,  is  the  bringing  of 
them  out  of  one  form  into  another,  in  order  to  prepare 
them  for  Addition,  Subtraction,  Multiplication,  &c. 

CASE    I. 

To  reduce  a  fraction  to  its  lowest  terms. 

RULE. 

Divide  the  greater  term  by  the  less,  and  that  divisor 
by  the  remainder,  and  so  continue  till  nothing  be  left;; 

*  The  following  perfect  numbers  are  all  which  are,  at  present,  known1, 

6  8589869056 

28  137438691328 

496  2305843008139952128 

8128  24178521639228158837784576 

33550336  9903520314282971830448816128 


80  REDUCTION   OP   VtfLGAB    FRACTIONS. 

the  last  divisor  will  be  the  common  measure;  then  divide 
both  parts  of  the  fraction  by  the  common  measure,  and 
the  quotients  will  express  the  fraction  required, 
c  Note. — If  the  common  measure  happen  to  be  1,  the 
fraction  is  already  in  its  lowest  term.  Cyphers,  on  the 
right  hand  side  of  both  terms,  may  be  rejected ;  as  ||£  ? • 

FXAMPLES. 

1.  Reduce  if  4.  to  its  lowest  terms. 

144)240(1  48)144(3 

144  144 

96)144(1  I  Ans, 

96  48)240(5 

Greatest  common  —  240 

measure  48)96(2 
96 

Thus  48  is  the  greatest  common  measure,  and  the 
true  answer  is  obtained  by  dividing  the  fraction  by  it. 

This  reduction  may  be  performed,  also,  by  another 
rule,  thus: — Divide  the  numerator  and  denominator  of 
the  fraction  by  any  number  that  will  divide  them  both 
without  a  remainder;  divide  the  quotients  in  the  same 
manner,  and  so  on,  'till  no  ir.tmber  will  divide  them  both? 
and  the  last  quotients  will  express  the  fraction  in  its 
lowest  terms. 

The  same  sum  done  by  this  method : — 

12)m(i*    '     4>2L!(f  Answer. 

2.  Reduce  -f^  to  its  lowest  terms.  Ans.  J. 

3.  Reduce  iff  to  its  lowest  terms.  Ans.  -i. 

4.  Reduce  |f  |  to  its  lowest  term?.  Ans.  T^. 

5.  Reduce  }Jf£  to  its  lowest  terms.  Ans.  }. 

CASE    II. 

To  reduce  a  mixed  number  to  an  improper  fraction. 

RULE. 

Multiply  the  whole  number  by  the  denominator  of 
the  fraction,  and  add  the  numerator  to  the  product;  then 
set  that  sum,  namely,  the  whole  product,  above  the  de- 
nominator for  the  fraction  required. 


REDUCTION  OF  VULGAR  TRACTIONS.       &1 

EXAMPLES. 

1.  Reduce  23f  to  an  improper  fraction. 
5 

117  '  y  Answer. 

2.  Reduce  12  J  to  an  improper  fraction.  Ans.  1%S, 

3.  Reduce  14  T7¥  to  an  improper  fraction.      Ans.  l£?. 

4.  Reduce  1 03  -/j-  to  an  improper  fraction.  Ans.  3|f 8. 

CASE    III. 

To  reduce  an  improper  fraction  to  a  whole  or  mixed  num- 
ber. 
RULE. 

Divide  the  numerator  by  the  denominator,  and  the 
quotient  will  be  the  whole  or  mixed  number  sought. 
EXAMPLES. 

1.  Reduce  ^   to  its  equivalent  number. 

3)12(4  Answer. 
12 

2.  Reduce  V5  to  its  equivalent  number. 

7)15(21  Answer. 
14 

1 

3.  Reduce  7T4T9  to  its  equivalent  number.    Ans.  44  y1^ 

4.  Reduce  5T6  to  its  equivalent  fraction.  Ans.  8. 

5.  Reduce  'f| 2  to  its  equivalent  fraction.    Ans.  54  \\. 

6.  Reduce  9Ty  to  its  equivalent  number.    Ans.  171  T|. 

CASE    IV. 

To  reduce  a  whole  number  to  an  equivalent  fraction,  having 
a  given  denominator. 

RULE. 

Multiply  the  whole  number  by  the  given  denomina- 
tor; then  set  the  product  above  for  a  numerator,  and 
the  given  denominator  below,  and  they  will  form  the 
fraction  required. 

EXAMPLES. 

1.  Reduce  9  to  a  fraction  whose  denominator  shall 
be  7.  9X7=63,  then  6T3  is  the  answer, 

2  Reduce  13  to  a  fraction  whose  denominator  shall 
be  12.  Ans,  ^ . 


82       REDUCTION  OF  VULGAR  FRACTION^. 

3.  Reduce  27  to  a  fraction  whose  denominator  shaU 
be  11.  Ans.  Yy7- 

CASE   V. 
To  reduce  a  compound  fraction  to  a  simple  one. 

RULE. 

Multiply  all  the  numerators  together  for  a  new  nu 
merator,  and  all  the  denominators  for  a  new  denomina* 
tor?  then  reduce  the  fraction  to  its  lowest  term. 

EXAMPLES. 

1.  Reduce  \  of  f  of  f  to  a  single  or  simple  fraction-. 

1X2X3          6  1 

..    ._    __  _.  —   _.  _  Answer. 

2X3X4          24          4 

2.  Reduce  |  of  f  of  \\  to  a  single  fraction.    Ans.  T4T. 

3.  Reduce  ^  of  £  to  a  single  fraction.  Ans.  ||. 

4.  Reduce  f  of  A  Of  \\  to  a  simple  fraction.    Ans.  ££$. 

CASE    VI. 

To   reduce  fractions  of  different  denominations  to  others 
of  the  same  value,  and  having  a  common  denominator. 

RULE. 

Multiply  each  numerator  into  all  the  denominators  ex-' 
eept  its  own,  for  a  new  numerator,  and  all  the  denomi^ 
nators  into  each  other  for  a  common  denominator.* 

EXAMPLES. 

1.  Reduce  i,  f,  and  f  to  a  common  denominator. 
1X3X4  =  12  the  numerator  for  |. 
2X2X4=16  the  numerator  for  f. 
3X2X3  =  18  the  numerator  for  f. 

*The  least  common  denominator,  or  multiple,  of  two  or  more  numbers,  may  be 
found  thus:  Divide  the  given  denominators  by  any  number  that  will  divide  two  or 
more  of  them  without  a  remainder,  and  set  the  quotients  and  undivided  numbers 
underneath.  Divide  these  quotients  by  any  number  that  will  divide  two  or  more 
of  them  as  before,  and  thus  continue,  'till  no  two  numbers  are  left,  capable  of  be- 
ing lessened.  Then  multiply  the  last  quotients,  and  the  divisor,  or  divisors  together, 
tiii'l  'li"  product  will  be  the  answer. 

What  is  the  least  common  multiple  of  f,  J,  TGT,  and  T^? 
8)9     8     15     16 


3)9     1      15       2 

3152 

3X1X5X2=30X3X8=720,  Ans. 


REDUCTION"  OF  VULGAR  TRACTIONS.       83 

Deneminator  2X3X4=24  the  common  denominator. 
Therefore,  the  results  are  ij,  |f  and  i|. 

Or  the  multiplications  may  be  performed  mentally, 
and  the  results  given  -'-,  |,  &  =i|,  if,  i|. 

2.  Reduce  f  and  f  to  a  common  denominator. 

Ans.  |f  and  f  f . 

3.  Reduce  f ,  f ,  and  J  to  a  common  denominator. 

Ans.  |£,  |f,  |f. 

4.  Reduce  £,  f  and  J  to  fractions  of  a  common  denomi- 
nator. Ans   ,yL,  Jfo  and  }££, 

CASE   VII. 

To  reduce  a  fraction  of  one  denomination  to  the  fraction 
of  another,  but  greater,  retaining  the  same  value. 

RULE. 

Make  the  fraction  a  compound  one,  by  comparing  it 
with  all  the  denominations  between  it  and  that  denomi- 
nation to  which  you  would  reduce  it;  then  reduce  that 
compound  fraction  to  a  simple  one. 

EXAMPLES. 

1.  Reduce  4  of  a  cent  to  the  fraction  of  a  dollar.    By 
comparing  it,  it  becomes  4  °f  yV  °f  TO*  which  being  re^ 
duced  by  case  five,  will  be  4X1X1=4  and  7X10X10 
=700.  Ans.  ^  D. 

2.  Reduce  |  of  a  mill  to  the  fraction  of  an  eagle. 

Ans.  jof  o-o- 

3.  Reduce  f  of  a  penny  to  the  fraction  of  a  pound. 

3X1  X  1  =    3  1 

|0fTVof¥V        -     —     —     =£ Ans' 

5X12X20=1200         400. 

4.  Reduce  -|  of  an  ounce  to  the  fraction  of  a  pound, 
Avoirdupois  Weight.  Ans.  JT  Ib, 

5.  Reduce  |  of  a  dwt.  .to  the  fraction  of  a  pound, 
Troy  Weight.  Ans.  T/^  Ib, 

6.  Reduce  }|  of  a  minute  to  the  fraction  of  a  day. 

Ans.  TTV?  day. 

CASE   VIII. 

To  reduce  the  fraction  of  one  denomination  to  the  fraction 
of  another,  but  less,  retaining  the  same  value. 


84        REDUCTION  OF  VULGAR  FRACTieNS. 
RULE. 

Multiply  the  given  numerator  by  the  parts  in  the  de- 
nomination between  it  and  that  to  which  you  would  re- 
duce it,  and  place  the  product  over  the  given  denomi- 
nator. 

EXAMPLES. 

1.  Reduce  ,^j  of  a  dollar  to  the  fraction  of  a  cent. 
The  fraction  is  ,^  of  V°  of  V;  then, 

J_X10XU)_100  and  this  reduced,  is  equal  to 
175X  ]  X  J  ~~175  Ans.  4  c. 

2.  Reduce  T^£  77  of  an  eagle  to  the  fraction  of  a  mill 

Ans.  |. 

3.  Reduce  ^i^-  of  a  pound  to  the  fraction  of  a  penny. 

An?,  f . 

4.  Reduce  ^  of  a  pound  Avoirdupois,  to  the  fraction 
of  an  ounce.  Ans.  £. 

5.  Reduce  l¥7^  of  a  pound  Troy,  to  the  fraction  of  a 
pennyweight.  Ans.  £  dwt. 

6.  Reduce  ,  JJ¥  of  a  day  to  the  fraction  of  a  minute. 

Ans.  |£  of  a  min. 

CASE   IX. 

To  find  the  value  of  the  fraction  in  the  known  parts  of  the 
integer;  or,  to  reduce  a  fraction  to  its  proper  value. 

RULE. 

Multiply  the  numerator  by  the  known  parts  of  the  in- 
teger, end  divide  by  the  denominator. 

EXAMPLES. 

1.  What  is  the  value  of  |  of  a  pound? 
2  thirds  of  a  pound. 
20 

3)40  thirds  of  a  shilling. 

13s.+l   third  of  a  shilling. 
12 

3)12  thirds  of  a  penny. 
4d.  An8.  13s.  4d. 


REDUCTION  OF  VULGAR  FRACTIONS.       85 

5.  Reduce  f  of  a  shilling  to  its  proper  value 

2  fifths  of  a  shilling. 
12 

5)24(4d. 
20 

4  fifths  of  a  penny. 
4 

6)16  fifths  of  a  farthing. 

3  qr.-f  1  fifth.  Ans.  4d.  3qr.  £. 

3.  Reduce  |  of  a  Ib.  Troy,  to  its  proper  quantity. 

Ans.  7  oz.  4dwt 

4.  Reduce  |  of  a  mile  to  its  proper  quantity. 

Ans.  6  fur.  16  poles. 

5.  Reduce  T5^  of  a  cwt.  to  its  proper  quantity. 

Ans.  2  qrs. 

6.  Reduce  f  0f  an  acre  to  its  proper  value. 

Ans.  2R.  20P. 

7.  Reduce  fV  of  a  day  to  its  proper  value. 

Ans.  7  hours  12  min, 

CASE    X. 

To  reduce  any  given  quantity  to  the  fraction  of  a  greater 
denomination  of  the  same  kind. 

RULE. 

Reduce  the  given  quantity  to  the  lowest  denomina- 
tion mentioned  for  a  numerator,  and  the  integer  into  the 
same  denomination  for  a  denominator. 

EXAMPLES. 

1.  Reduce  16s.  8d.  to  the  fraction  of  a  pound; 
16     8  Integer  £1 

12  20 

Numerator    200  20 

=     £  Ans.  !«-- 

Deno«unator240  

— r  240  Denominator^ 
8 


ot>  ADDITION    OF    VULGAR    FRACTIONS. 

2.  Reduce  6  furlongs  and  16  poles  to  the  fraction  ot 

a  mile.  Ans.  £ . 

3.  Reduce  f  of  a  farthing  to  the  fraction  of  a  pound. 

Ans.  TT'¥Tr. 

4.  Reduce  £  dwt.  to  the  fraction  of  a  pound  Troy. 

Ans.  ^. 

5.  Bring  80  cents  to  the  fraction  of  a  dollar. 

A  dollar  is  100  cents,  then  80  cents  are  equal  to  T8/ff 
of  a  dollar;  which,  being  reduced,  is  equal  to  f  Ans. 

6.  Bring  16  cents  9  mills  to  the  fraction  of  an  eagle, 

16  cents  9  mills  =       169 


1  eagle  =  10000 

7.  Bring  2  quarters  3£  nails  to  the  fraction  of  an  ejl< 
English.* 

2  quarters  3i  nails. 
4 

11 
•9 

Numerator       100 

Denominator       9  of  -J  of  J  =  ||£  =  A  Ans. 


ADDITION   OF    VULGAR    TRACTIONS. 

CASE    I. 
To  add  fractions  having  a  common  denominator. 

PtULE. 

Add  all  the  numerators  together,  and  place  the  sum 
over  the  common  denominator,  which  will  give  the  sum 
of  the  fractions  required. 

EXAMPLES. 
1.  Add  f ,  f  and  f  together. 

!+!+!  =  *=  H  Answer. 

*  When  the  sum  contains  a  fraction,  as  in  the  7th  example,  multiply  both  p.-uts 
ot  the  sum  by  the  denominator  thereof,  and  to  the  numerator  add  the  nunn  r-itoi 
of  the  given  fraction. 


ADDITION  OF  VULGAR  TRACTIONS.        87 

&.  Add  j,  5,  %  and  -f  together. 

'  Y+f+f+f  ='  V  =  H  Answer. 

CASE    II. 
7o  add  fractions  having  different  denominators. 

RULE. 

Find  the  common  denominator  by  Case  VI,  in  Reduce 
lion;  then  add,  as  in  the  preceding  example?. 

EXAMPLES. 

1.  Add  |  and  f  together. 

4X5=  20) 

•   numerators. 


47  sum. 

4X9  =  36  com.  denom.  f  J  ±=  l£.i  Ans. 
2.  Add  f  a«d  T5T  together.  Ans.  1§r. 

CASE    III. 
To  add  mixed  numbers. 

RULE. 

Add  the  fractions  as  in  Case  I,  in  Addition,  and  the 
whole  numbers  as  in  Simple  Addition;  then  add  the  frac- 
tions to  the  sum  of  the  whole  numbers.  If  the  fractions 
have  different  denominators,  reduce  them  to  a  common 
denominator,  and  then  add  the  fractions  to  the  integers 
or  whole  numbers. 

EXAMPLES. 

1.  Add  13TL,  9T45  and  3T^  together. 

13-f9-f-3  =  25  whole  numbers. 

TV+T45+T7s  =  H  =  f      Thus>  25f  Ans. 

2.  Add  5|,  6f  and  4i  together. 

5-J-6+4  =15  whole  numbers. 
Then,   2X8X2  =  32 
7X3X2  =  42 
1X3X8=  24 

98  sum  of  the  numerators. 

3X8X2  =  48  common  denominator. 
Then,  f|  =  2^.         Thus,  15+2^  =  17 JT  Answer, 
''}.  Add  If,  2}  and  3f  together.  Ans.  7}$& 


38  ADDITION    OF   VULGAR 


CASE    IV. 

To-  add  compound  fractions. 

RULE. 
Reduce  them  to  simple  ones,  and  proceed  as  before 

EXAMPLES. 

1.  Add  i  of  I  of  £,  to  I  of  J  of  }f 


— 
2X3X4  =  ^^  simple  fraction. 

2  X3  X  10  — 
and  3X5XH  =  A°5  =  T4i  simple  fraction. 

Then  find  a  common  denominator. 

-  .,        4X   4=  16 

for  J,  ^  thus,  i  x  j  j  _  ||  numerator. 

27  sum  of  the  nuirieratop$. 

4XH  =  44  common  denominator. 

Therefore  f  J  is  the  Answer, 
2.  Add  -f.  9i  and  |  of  i  together. 
jVo^e>  —  The  mixed  number  of  9J  «=  4/  ;  the  compound 
fraction  f  of  -|-  •=  f.     Then  the  fractions  are,  f,  y  and 
|;  which  must  be  reduced  to.  fractions  of  a  common  de- 
nominator and  added.  Ans. 


CASE   V. 

When  the  given  fractions  are  of  several  denominations. 

RULE. 

Reduce  them  to  their  proper  values,  or  quantities, 
and  add  them  according  to  the  following  examples, 

EXAMPLES. 

1.  Add  |  of  a  pound  to  |  of  a  shilling. 
Thus,  |  of  a  pound  =  13s.     4d. 
and  f  of  a  shilling  =    Os.     4d. 


13s.     8d.     SJqr. 

2.  Add  £  of  a  pound  and  ,3-  of  a  shilling  together. 

Ans.  15s.  10  ,^d. 

3.  Add  i  of  a  week.  -]-  of  a  day,  and  ^  of  an  hour  to- 
gether, Ans.  2d. 


SUBTRACTION    OF   VULGAR    FRACTIONS.  89 

4.  Add  £  of  a  yard,  J  of  a  foot,  and  £  of  a  mile  to- 
gether. Ans.  1100yds.  2ft.  7in. 

5.  Add  J  of  a  dollar,  |  of  a  cent,  ,3¥  of  a  cent,  and,  J 
of  a  mill  together.  Ans.  20c.  Urn. 

6.  Add  J  of  pound,  ^  of  a  shilling,  and  £  of  a  penny 
together.  Ans.  2s. 


SUBTRACTION    OF   VULGAR   FRACTIONS', 
CASE   I. 

When  the  fractions  have  a  common  denominator. 

RULE. 

Subtract  the  less  numerator  from  the  greater,  and  se! 
the  remainder  over  the  common  denominator,  which 
will  show  the  difference  of  the  given  fractions, 

EXAMPLES. 

1.  Subtract  f  from  f.  Ans.  f 

2.  What  is  the  difference  between  f  and  f  .  Ans.  f  ={•, 

3.  Take  ,\  from  ^  •  Ans.  &==£. 

4.  Take  f  from  f  Ans.  f=i 

CASE   II. 

When  fractions,  or  mixed  numbers,  are  to  be  subtracted 
from  whole  numbers. 

RULE. 

Subtract  the  numerator  from  its  denominator,  and  un- 
der the  remainder  place  the  denominator;  then  carry 
One  to  be  deducted  from  the  whole  number. 

EXAMPLES. 
J.  Take  |  from  12. 

Thus,     12 


llf  Answer, 

2.  Subtract  27f$  from  32.  Ans.  4J$  , 

3.  From  10,  take  TV  Ans.  9&\ 

4.  From  9,  take  5i.  Ans.  3£. 

5.  From  26,  take  24TV  Ans.  T\=  J-, 


90  SUBTRACTION    OF    VULGAR    FRACTIONS, 

CASE    III. 

To  subtract  fractions  having  different  denominators. 
RULE, 

Reduce  the  fractions  to  a  common  denominator,  by 
Case  VI  in  Reduction,  and  subtract  the  less  numerator 
from  the  greater — the  difference  will  be  the  answer. 

EXAMPLES. 

1.  What  is  the  difference  between  ij-  and  ff  ? 

Thus,  -J-a.  and  f  f  are  equal  to  |£i,  TW, 

And  88  from  171,  leaves  85.  Ans.  TW- 

2.  From  ^  take  f .  Ans.  JT. 

3.  Take  JL  from  f .  Ans.  |. 

4.  Subtract  T7^  from  ^.  Ans.  ^ 

CASE    IV. 

To  distinguish  the  largest  of  any  two  fractions. 

RULE. 

Reduce  them  to  a  common  denominator,  and  the  one 
that  has  the  larger  numerator  is  the  larger  fraction. 

EXAMPLE. 

Which  is  the  greater  fraction,  }J,  or  ff  ? 
Thus,  192  common  denominator. 

12X15  =  180  numerator. 
16X11=  H6  numerator. 

4  numerator. 

Then,  Tf  2  =  TV 
Therefore,  if  is  the  greater  fraction  by  ?L.,  Ans 

CASE    V. 

To  subtract  one  mixed  number  from  another,  when  thefra& 
tion  to  be  subtracted  is  greater  than  that  from  which  the 
subtraction  is  to  be  made. 

RULE. 

Reduce  the  fractions  to  a  common  denominator;  sub- 
tract the  numerator  of  the  greater  from  the  common 
denominator,  and  add  to  the  remainder  the  less  numera- 
tor; then  set  the  sura  of  them  over  the  common  denomi- 
nator, and  carry  one  to  the  whole  number,  and  subtract 
as'io  Simple  Subtraction. 


MULTIPLICATION    OF    VULGAR    FRACTIONS,  91 

EXAMPLES. 

1.  From  12£  subtract  8}f. 

Thus.  |  reduced  to  a  common  denominator,  =  T5T7?, 
and  ||  reduced  to  a  common  denominator,  =  y7^-. 

Then  72  taken  from  1  14,  leaves  42;  which,  added  to 

57,  the  less  numerator,  makes  99  for  the  numerator  in 

the  answer.     Then   carrying  1   to  the  whole  number, 

namely,  8  makes  it  9;  and    taking  9  from   12  leaves  3. 

Therefore,  the  answer  is 

2.  From  10^,  take  1TV  Ans. 

CASE    VI. 

When  fractions  are  of  different  denominations. 

RULE. 

Reduce  them  to  their  proper  values,  or  quantities, 
and  subtract  as  in  Compound  Subtraction,. 

EXAMPLES. 

1.  From  f  of  a  pound,  take  ^  of  a  shilling. 
Thus,  |  of  a  pound  =  17s.     6d. 
And  |  of  a  shilling  =0       4 


17s.     2d.  Answer. 


2.  From  f  of  a  ton  take  T97  of  a  cwt. 

Ans.  14cwt.  Oqr.  lllb.  3oz.  3}dr, 

3.  From  J  of  a  pound,  take  £  of  a  shilling,  and  what 
will  be  the  remainder?  Ans.  14s.  3d. 

4.  From  |  of  a  pound,  Troy  Weight,  take  £  of  an 
ounce.  Ans.  8oz.  16dwt.  16gr, 


MULTIPLICATION   OF   VULGAR   FRACTIONS. 

RULE. 

Reduce  compound  fractions  to  simple  ones,  and  mixed 
numbers  to  equivalent  fractions;  then  multiply  all  the 
numerators  together  for  a  numerator,  and  all  the  de- 
nominators together  for  a  denominator  which  will  give 
the  product  required, 


92  DIVISION   OF   VULGAR   FRAGTIO^ 

EXAMPLES, 

1.  Multiply  f  by  f. 

Here,  f  Xf  =  A  =  £»  the  answer. 

2.  Multiply  f  by  f.  Aus.  ^. 

3.  "  A  by  A-  Ans.Ty 

4.  "  £  of  7  byf  Ans.  If. 

5.  "  6|  by  f  Ans.  ||. 

6.  «  414.  by  3f .  Ans.  14i|^ 

7.  <{  41  byf  Ans.T%, 


DIVISION   OF   VULGAR   FRACTIONS. 

RULE. 

Reduce  compound  fractions  to  simple  ones,  and  mixed 
numbers  to  equivalent  fractions;  tben  multiply  the  nu- 
merator of  the  dividend  by  the  denominator  of  the  di- 
visor, for  a  new  numerator,  and  the  denominator  of  the 
dividend  by  the  numerator  of  the  divisor,  for  the  de- 
nominator; the  fractions  thus  formed  will  be  the  answer' 

EXAMPLES. 

1.  Divide  4  by  f. 

Thus,  4  numerator  of  the  dividend. 
3  X  denominator  of  the  divisor, 

12  numerator. 

Then  7  denominator  of  the  dividend-, 
2  X  numerator  of  the  divisor. 

14  denominator. 

Therefore,  }|  =  &  is  the  answer^ 

2.  Divide  f  by  f  . 

Thus, 
' 


A 
9  Answer, 

3.  Divide  Jf  by  f  Ans.  fv 


5.  "  |-  by  y.  Ans.  TV 

6.  «  Hbyf  AnM* 

7.  "  f  by  f  Ans.  -if 
S.  «            ^^ 


DECIMAL    TRACTIONS.  93 

9.  Divide  f  by  £.  Ans.  ^ . 

10.  "       71  by  9£»  Ans.  ff . 

11.  "       |  of  i  by  -f  of  7f .  Ans.  Tfp 

12.  What  part  of  33  JT,  is  28J--J.  Ans.  f . 

Q.  1.  What  are  Vulgar  Fractions? 

2.  How  are  they  represented  in  figures'? 

3.  What  is  the  upper  figure  called?- 

4.  What  is  the  lower  figure  called? 

5.  What  does  the  denominator  show? 

6.  What  does  the  numerator  show? 

7.  What  are  the  two  numbers  of  a  fraction  some** 

times  called? 


DECIMAL  FRACTIONS. 

Decimal  Fractions  are  parts  of  whole  numbers,  and 
lire  separated  from  them  by  a  point,  thus,  8.5;  which  is 
read,  eight  and  five  tenths,  or  8  T5^.  All  the  figures  on 
the  left  of  the  point  are  whole  numbers;  those  on  the 
tight  are  fractions.  An  unit  is  supposed  to  be  divided 
into  ten  equal  parts,  and  the  figure  at  the  right  of  the 
point  expresses  the  number  of  those  parts.  Decimals 
decrease  in  a  tenfold  proportion,  as  they  depart  from 
the  separating  point.  Thus,  .5  is  5  tenths,  or  one  half; 
.57  is  57  hundredths;  .05  is  5  hundredths;  and  .005  is  B 
thousandths.  Cyphers  placed  at  the  right  hand  of  de- 
cimals do  not  alter  their  value;  thus,  .5  or  T\;  .50  or 
To\i  -500  or  T5oW?  are  a^  °^  t*16  same  value,  and  equal 
to  |.  The  first  place  of  decimals  is  called  tenths;  the 
second,  hundredths,  &c. 

DECIMAL  JVtIMERATION   TABLE. 


1  1 


765    4    321     .     654321 


94  ADDITION  or  DECIMALS. 

ADDITION    OF    DECIMALS. 

RULE. 

Place  the  figures  according  to  their  values — units  un- 
der units,  tenths  under  tenths,  &c.,  and  add  as  in  Simple 
Addition  of  whole  numbers;  observing  to  place  the 
point  in  the  sum  under  those  in  the  given  numbers. 

EXAMPLES. 

1.  Add  together  the  following  sums,  viz:  252. 25? 
343.5,  17.85,  1244,75  and  .425. 

Thus,      252.25         Ab?e.—The  answer  to  this  sum 

343.5       is  read  thus:   One  thousand  eight 

17.85     hundred  and  fifty-eight,  and  seven 

1244.75     hundred    and   seventy -five   thou' 

.425  sandths. 

1858.775  Answer, 

ii.  in. 

87654.321  987654.3 

23456.78  212345.67 

98765.4  898765.432 


209876.501  2098765.402 


4.  Add  420.4,  38.05,  54.9,  27.003  and  29.384. 

Ans.  569.737. 

5.  Add  376.25,  86.125,  6.5,  41.02  and  358.865. 

Ans.  868.760. 

6.  Add  .64,  .840.  .4,  .04,  .742,  .86,  .99  and  .450. 

Ans.  4.962. 

Note. — Dimes,  cents  and  mills  are  decimals  of  a  dol- 
lar. A  dime  is  one  tenth,  a  cent  is  one  hundredth,  a 
mill  is  one  thousandth;  which  shows  that  the  addition 
of  Federal  Money  is  the  addition  of  decimals.  Thus, 
5  tenths  of  a  dollar  is  the  same  as  50  hundredths,  or  50 
cents;  and  25  hundredths  of  a  dollar  is  equal  to  25 
cents,  &c.  It  may  he  likewise  added,  that  .5,  or  .60,  or 
.500,  being  equal  to  one  half,  .25  equal  to  one  quarter, 
and  .75  equal  to  three  quarters  or  three  fourths,  so  .7, 
or  .35,  or  any  intermediate  fractions,  have  a  propor- 
tionate value. 


SUBTRACTION  AND  MULTIPLICATION  OF  DECIMALS. 


SUBTRACTION    OF    DECIMALS. 

RULE. 

Write  the  larger  number  first,  and  the  smaller  one 
under  it;  then  subtract  as  in  Simple  Subtraction;  obser- 
ving, that  the  dividing  point  in  the  answer,  or  remain 
der,  must  be  placed  under  those  in  the  sum. 

EXAMPLES. 


From 
take 


i. 
91.73 

2.138 


n. 

2.73 
1.9185 


89.592 


0.8115 


in. 

214.81 
4.90142 

209.90858 


IV. 

From    1.5 
take      .987654321 


0.512345679 


v. 

.8234567890 
.5987654329 

.2246913561 


MULTIPLICATION    OF    DECIMALS. 

RULE. 

Place  the  multiplier  under  the  multiplicand,  and  mul- 
tiply as  in  Simple  Multiplication;  then  point  off  as  many 
places  for  decimals  as  there  are  decimals  in  the  multi- 
plicand and  multiplier.  If  there  be  not  so  many  figures 
in  the  product  as  there  are  decimals  in  both  factors,  the 
deficiency  must  be  supplied  by  prefixing  cyphers. 
EXAMPLES. 


Multiply  24.85 
by     6.25 

12425 
4970 
14910 


155.3125 


228375 
3197*5 
365400 

3996.5625 


in. 

79.347 
23.15 

396735 
79347 
238041 
158694 

1836.88305 


96  DIVISION   OF  DECIMALS. 

IV.                    V.                         VI.  VII. 

Multiply  .63478         .567               .285  .25 

by      .8994              5               .003  .25 


253912  2.835         .000855  125 

571302 50 

571302 
507824  .0625 


.570921132 


£.  Multiply  .63478  by  .8204.  Ans.  .520773512. 

'9.  Multiply  .385746  by  .00464.     Ans.  .00178986144. 


DIVISION    OF   DECIMALS. 

RULE. 

Divide  as  in  Simple  Division,  and  point  off  as  many 
ugures  from  the  right  hand  of  the  quotient,  for  decimals, 
as  the  decimal  figures  in  the  dividend  exceed  in  number 
those  in  the  divisor.  When  there  are  not  so  many  fi- 
gures in  the  quotient  as  this  rule  requires,  the  deficiency 
must  be  supplied  by  prefixing  cyphers  to  the  left  of  the 
quotient.  When  there  are  more  decimal  figures  in  the 
divisor  than  in  the  dividend,  place  as  many  cyphers  to 
the  right  of  the  dividend  as  will  make  them  equal.— 
When  the  number  of  decimals  in  the  divisor,  and  the 
number  in  the  dividend  are  equal,  the  quotient  will  al- 
ways be  in  whole  numbers,  unless  there  should  be  a  re- 
mainder after  the  dividend  is  all  brought  down.  When 
there  is  a  remainder,  cyphers  must  be  annexed  to  it 
and  the  division  continued  and  the  quotient  thence  ari- 
sing will  be  decimals. 


^DIVISION   OP   DECIMALS. 

EXAMPLES. 
II. 


6.4)128.64(20.1  324.8)9876.5(30.4079 
128  9744 


64 
64 


IV. 

,5).75(1.5 
5 

25«. 
25 


13250  cypher 
12992  annexed. 


25800 
22736 

30640 
29232 


1408+ 


in. 

.48)65.88(137 
48 

178 
144 


348 
336 

12rem 


v. 


VI. 


179).48624097(.0027I643 
358 


1282 
1253 


.2685)27.0000(100.55865 
26.85 


15000  cyphers 
13425  annexed. 


294 
179 

1150 
1074 


35750 
13425 

23250 
21480 


769 
716 

537 
537 


7.  Divide  234.70525  by  64.25. 

8.  Divide  14  by  .7854. 

8.  Divide  2175.68  by  100. 
9 


17700 
16110 

15900 
13425 

2475 


Ans.  3,653 

Ans.  17.825. 

Ans.  21.7568. 


98  REDUCTION    OF    DECIMALS* 

REDUCTION    OF   DECIMALS. 
CASE    I. 

To  reduce  a  vulgar  fraction  to  a  decimal. 

RULE. 

Place  cyphers  to  the  right  of  the  numerator  until  you 
can  divide  it  by  the  denominator;  and  divide  'till  noth- 
ing remains;  or,  if  it  be  a  number  that  will  not  divide 
without  a  remainder,  then  divide  until  you  get  three  or 
more  figures  for  the  quotient.  The  quotient  will  be  thf, 
vulgar  fraction  expressed  in  decimals. 

EXAMPLE^. 

1.  Reduce  J  to  a  decimal. 

Thus,  2)1.0(.5 
10 

2.  Reduce  \  and  |  to  decimals. 

4)1.00(.24  4)S.OO(.75 

8  28 

20  20 

20  20 

3.  Reduce  £  to  a  decimal. 

3)1.0Q(.333  Answer, 


10 

9 

10 
9 


4.  Reduce  £  to  a  decimal.  Ans.  .376. 

CASE    II. 

To  reduce  any  sum,  or  quantity,  to  the  decimal  of  any 
given  denomination. 


REDUCTION    OP    DECIMALS.  9§ 

RULE. 

Reduce  the  quantity  to  the  lowest  denomination,  and 
reduce  the  proposed  integer  to  the  same  denomination; 
then  divide  the  quantity  by  the  amount  of  the  integer, 
and  the  quotient  will  be  the  answer. 

EXAMPLES. 

1.  Reduce  3s.  9d.  to  the  fraction  of  a  pound. 
One  pound  reduced  to  pence  makes  240;  and  3s.  9dr 
reduced  to  pence  makes  45. 

Then,  240)45.0000(.1875  Answer: 
240 

2100 
1920 

1800 
1680 


1200 
1200 

The  same  sum  may  be  done  by  writing  the  given  num- 
bers from  the  least  to  the  greatest  in  a  perpendicular 
column,  and  dividing  each  of  them  by  such  number  as 
tvill  reduce  it  to  the  next  denomination,  annexing  the 
quotient  to  the  succeeding  number. 
Thus— 

12     9.00 
3.750|0 

.1875  Answe3r. 

2.  Reduce  7  drams  to  the  decimal  of  a  pound,  Avoir- 
dupois Weight.  Ans.  .02734375. 

3.  Reduce  14  minutes  to  the  decimal  of  a  day. 

Ans.  .009722. 

4.  Reduce  21  pints  to  the  decimal  of  a  peck. 

Ans.  .013125. 

5.  Reduce  15s.  6d.  to  the  decimal  of  a  pound. 

Ans.  .775. 

6.  Reduce  56  gallons  3  quarts  1  pint  to  the  decimal 
of  a  hogshead.  Ans.  .9027: 


IOQ  REDUCTION    OF   DECIMALS. 

7.  Reduce  12dwts.  16grs.  to  the  decimal  of  a  poand>, 
Troy  Weight.  Ans,  .0527- 

8.  Reduce  4  mills  to  the  decimal  of  a  dollar  Ans,  .004. 

9.  Reduce  7  cents  to  the  fraction  of  a  dollar.  Ans.  .07= 
Note. — In  doing  sums  in  this  rule,  it  will  be  necessary 

to  keep  in  mind  the  tables  of  the  different  weights,, 
measures,  money,  &c. 

CASE   III.. 
To  find  the  value  of  any  decimal  fraction. 

RULE. 

Multiply  the  decimal  by  the  number  of  parts  in  the 
next  lower  denomination;  point  off  as  many  figures  for 
decimals  as  is  required  by  the  rule  in  nnulti plication  of 
decimals;  then  multiply  the  decimal  by  the  number  of 
parts  in  the  next  lower  denomination,  and  so  on,  to  the 
last.  The  figures  on  the  left  of  the  points  will  show  the 
value  of  the  decimal  in  the  different  denominations. 

EXAMPLES. 

L  What  is  the  value  of  .775  of  a  pound? 

£.776 
20 

5.15.500 
12 


d.6.000  Answer  15s.  6d. 

,  What  is  the  value  of  .625  of  a  cwt.i 
4 


4000 
1000 


14.000  Ans.  2qr.  14ib. 

3.  What  is  the  value  of  .625  of  a  shilling?   Ans.  7-J-d 

4.  What  is  the  value  of    .4694  of  a  pound,    Troy 
Weight?  Ans,  5oz,  12dwts,  I5,744gr?, 


DUODECIMALS.  101 

5.  What  is  the  value  of  .6875  of  a  yard  ?  Ans.  2qrs.  3na, 

6.  What  is  the  value  of  .3375  of  an  acre  ?  Ans.  1 R .  1 4P. 

7.  What  is  the  value  of  .0008  of  an  Eagle?  Ans.  8m, 
Q.  1.  What  are  decimal  Fractions? 

2.  How  are  they  separated  from  whole  numbers? 

3.  In  what  manner  do  they  decrease  as  they  depart 

from  the  separating  point? 

4.  In  the  table  of  numeration,  what  is  the  first  place 

called? 

5.  What  money,  or  currency,  is  reckoned  after  the 

manner  of  Decimal  fractions? 


DUODECIMALS. 

Duodecimals  are  fractions  of  a  foot  or  of  an  inch,  or 
parts  of  an  inch,  and  have  12  for  their  denominator. — 
They  are  useful  in  measuring'  planes,  or  surfaces,  and 
solids.  In  adding,  subtracting,  and  multiplying  by  Duo- 
decimals, it  is  necessary  to  carry  one  for  twelve. 

The  denominations  are  feet,  inches,  seconds,  third? 
and  fourths. 

12  fourths  ""         make         1  third 

12  thirds      -  1  second      ". 

12  seconds  -  1  inch          /. 

12  inches     -  1  foot          Ft, 


ADDITION   OF   DUODECIMALS, 

RULE. 

$5- Add  as  ip  Compound  Addition, 

I.  EXAMPLES.  II. 

Ft.  I.  "  '"  Ft.  I.  "  '"  "h 

24  10  11  10  '    80  11  %10  10  11 

18  9  8  3  25  4  3  2  1 

12  10  1  7  75  10  11  11  10 

56   6   9   8     182   3   2   0  10 
9* 


102  DUODECIMALS. 

SUBTRACTION   OF   DUODECIMAL^ 

RULE. 

(ffir  Subtract  as  in  Compound  Subtraction. 

EXAMPLES. 

1.  II. 

Ft.  I.  "  '"  Ft.  I.  "  '"  "" 

18   9   8  3  80  1  2  4  6 

12  10  11  1.0  39  11  10  10  8 


5  10   8   5     40   1   3   5  10 


MULTIPLICATION   OF  DUODECIMALS. 

RULE. 

Set  down  the  different  denominations,  one  under  the 
other,  so  that  feet  stand  under  feet,  inches  under  inches? 
seconds  under  seconds,  &c.  Multiply  each  denomina- 
tion in  the  sum,  by  the  feet  in  the  multiplier,  and  set  the 
result  of  each  under  its  corresponding  term,  observing 
to  carry  one  for  every  12  from  one  denomination  to 
another.  Then  multiply  the  sum  by  the  inches  in  the 
multiplier,  and  set  the  result  of  each  term  one  place 
removed  to  the  right  of  those  in  the  sum;  and  in  like 
manner,  multiply  the  sum  or  multiplicand  by  seconds^ 
fhirds,  &c.,  if  there  be  aoy  in  the  multiplier. 

Or,  instead  of  multiplying  by  inches,  &,c.,  take  such 
parts  in  the  multiplicand,  as  these  are  of  a  foot. 

Add  the  amount  of  the  multiplications  together,  and 
#ieir  sum  will  b*  the  answer. 

EXAMPLES. 
i. 
Ft.      I. 

Multiply  4         7 
by  6         4 

27         6 
1         6       4 

29         0       4"  66         4       6" 


DUODECIMALS* 


103 


Ft. 

Multiply  8 
by  4 

in. 
/.       " 

4       2 

2 

/// 
10 

IV. 

Ft.       I. 

11       10 
10         9 

33 
1 

4     11 

4        8 

4 
5     8 

118         4 
8       10 

34 


7       9     8""   127         2     6" 


JVote. — In  doing  the  third  sum,  I  begin  with  4,  which 
stands  under  the  8,  and  multiply  the  sum,  beginning  with 
the  right  hand  figure  which  is  10;  saying  4  times  10 
are  40.  In  40,  I  find  there  are  3  times  12  and  4  over. 
Setting  down  4, 1  multiply  the  next  figure,  adding  three 
to  it,  which  makes  11,  and  thus  multiply  the  whole  sum. 
Then  taking  the  2  for  the  multiplier,  I  say  2  times  10 
are  20.  In  20  I  find  12  is  contained  once,  and  8  over. 
Setting  down  8  one  place  farther  to  the  right,  I  say  2 
times  2  are  4,  and  one  to  carry  makes  5;  and  after  this 
manner  multiply  all  the  figures  in  the  sum.  Then  ad- 
ding the  two  rows  of  figures  together,  1  obtain  the  an- 
swer. 

Method  of  doing  the  same  sum  by  taking  the  frac? 
tional  parts. 

Ft.      I.       "     "' 
2  inches  = 


8""  Answef, 


In  this  last  example,  I  multiply  the  sum  by  4,  as  in  the 
former  case.  Then,  as  2  inches  make  i  of  a  foot,  I  di- 
vide the  Sjum  by  6,  which  I  had  multiplied  by  4,  divi- 
ding it  after  the  manner  of  Compound  Division,  multi- 
plying each  remainder  by  12,  and  adding  it  to  the  next 
lower  denomination;  and  setting  the  result  under  the 


1 

8 

4 

2 

10 

4 

2 

33 

4 

11 

4 

1 

4 

8 

5 

34 

9 

7 

9 

104  DUODECIMALS. 

amount  of  the  multiplication.     Then  I  add  the  two  sums 
as  before, 

5.  What  are  the  solid  contents  of  a  cubick  block  that 
is  4  feet  4  inches  in  length,  3  feet  8  inches  in  breadth, 
and  2  feet  8  inches  in  thickness? 

Ft.          L 

4  4 

3  8X 


13  0 

2  10     8 

15  10     8 

2  8X 

31  9     4 

10  714 

42  4     5"  4'"  Answer. 


6.  What  is  the  product  of  12  feet  9  inches,  multiplied 
by  6  feet  4  inches.  Ans.  SOFt.  91. 

7.  What  is  the  product  of  3  feet  2  inches  3"  multi- 
plied by  3  feet  2  inchesS"?  Ans.  lOFt.  II.  11"0'"  9"". 

8.  What  is  the  price  of  a  marble  slab,  whose  length 
is  5  feet  7  inches,  and  breadth  1  foot  10  inches,  at  one 
dollar  per  foot?  Ans.  $  10.23, 

Q.  1.  What  are  Duodecimals? 

2.  In  what  are  they  useful? 

3«  In  adding,  subtracting,  and  multiplying  Duodeci- 
mals, what  d©  you  observe  in  carrying  from 
one  denomination  to  another? 

4.  What  are  the  denominations  used  in  Duodecimals? 

5.  Repeat  the  rule  for  Multiplication  of  Duodecimals9. 


SINGLE   RULE   OF   THREE*  105 

SINGLE  RULE  OF  THREE. 

The  Rule  of  Three,  which  is  sometimes  called  the 
Rule  of  Proportion,  teaches  how  to  find  a  fourth  pro- 
portional to  three  numbers  given.  As  it  has  three  terms 
given  to  find  a  fourth,  it  is  generally  called  the  Rule  of 
Three. 

Questions  to  prepare  the  learner  for  this  rule. 

1.  If  2  apples  cost  3  cents,  how  much  will  4  applet 
cost  at  the  same  rate? 

2.  If  you  give  2  cents  for  4  nuts,  how  many  cents* 
must  you  give  for  8  nuts? 

3.  If  a  pound  of  butter  cost  8  cents,  how  much  will 
4  pounds  cost? 

4.  A  boy  has  20  melons  to  sell,  and  asks  10  cents  for 
two,  how  much  will  they  all  come  to  at  the  same  rate? 

5.  If  6  men  can  reap  a  field  of  wheat  in  4  days,  how 
long  will  it  take  12  men  to  reap  the  same  field? 

6.  If  4  yards  of  cloth  cost  1  dollar,  how  much  will  2 
yards  cost? 

7.  How  much  will  a  gallon  of  milk  come  to,  at  four 
cents  a  quart? 

8.  How  milch  will  a  bushel  of  peaches  come  to,  at 
25  cents  a  peck? 

9.  If  2  cents  will  buy  3  apples,  how  many  apples  will 
9  cents  buy? 

10.  If  a  boy  can  run  2  miles  in  one  hour,  how  far  can 
he  run  in  4  hour*? 

RULE. 

Set  the  term  in  the  third  place,  which  is  of  the  same 
kind  with  that  in  which  the  answer  is  required.  Then 
determine  \vhether  the  answer  ought  to  be  greater  or 
less  than  the  third  term.  If  the  answer  ought  to  be 
greater  than  the  third  term,  set  the  greater  of  the  other 
two  numbers  on  the  left  for  a  second  or  middle  term; 
and  the  less  number  on  the  left  of  the  second  term,  for 
a  first  term.  If  the  answer  ought  to  be  less  than  the. 
third  term,  the  less  of  the  two  other  numbers  must  be 
the  middle  term,  and  the  greater  must  be  the  first  ternu 


106  SINGLE    RULE   OF    THREE. 

After  thus  stating  the  sum,  proceed  to  do  it  in  the  fol- 
lowing manner,  viz  :  Reduce  the  third  term  to  the  lowest 
denomination  mentioned  in  it.  Reduce,  likewise,  the 
first  aod  second  terms  to  the  lowest  denomination  that 
either  of  them  hag.  Then  multiply  the  second  and 
third  terms  together,  and  divide  their  product  by  the 
first  term.  The  quotient  thus  obtained  will  be  the  an* 
swer. 

It  will  not  be  necessary  to  distinguish  between  direct 
and  inverse  proportion,  because  the  foregoing  rule  is 
calculated  for  both. 

PROOF. 

By  reversing  the  statement. 

EXAMPLES. 

1.  If  3  pounds  of  sugar  cost  2&  cents,  what  will  IB 
pounds  cost  at  the  same  rate? 

Ibs.     Ibs.         cts. 
Thus,    3   :   18   :    :  25 
18 

200 
25 

3)450 

$1.50  Answer. 

2.  If  7  pounds  of  coffee  cost  87|  cents,  what  must  f 
pay  for  244  pounds? 

Ibs.     Ibs.  cts. 

7    :  244    :    :  87 


1708 
1952 
122 


7)21350 
$30.50  Answer 


SINGLE    RULE    OF    THREE*  107 

3.  If  450  barrels  of  flour  cost  $  1350,  what  will  8  bar- 
rels cost? 

bbls.  bbls.          $          $ 
Thus— As  450   :  8   :    :   1350   :  24,  Answer:* 

4.  If  15  yards  of  cloth   cost   £6,  what  number  of 
yards  may  be  bought  for  £125? 

£      £  yds.      yds. 

As  6    :   125   :   :   15   :  312|  Answer. 

5.  If  12  men  can   do  a  piece  of  work  in  20  days,  ip 
what  time  will  18  men  do  it? 

m.      m.  d.       d. 

As   18    :   12    :    :  2Q   :   13|  Answer. 

6.  What  will  be  the  cost  of  17  tons  of  lead,  at  22£ 
dollars  66  cents  per  ton? 

T.     T.  D.  cts.        D.  cts. 

As   1    :   17   :    :  223.66    :  2802.22  Ans. 

7.  WTmt  will  72  yards  of  cloth  cost  at  the  rate  of  9 
yards  for  £5J2s. 

~yds.  yds.     £  s.       £     s. 
As  9    :  72    :    :  5   12    :  44   16  Ans, 

8.  If  750  men  require  22500  rations  of  bread  for  a 
month,  what  will  a  garrison  of  1200  require?  Ans,  36000. 

9.  What  must  be  the  length  of  a  board  that  is  9  inches 
in  width,  to  make  a  surface  of  144  inches,  or  a  square 
foot?  Ans.  16  inches. 

10.  How  many  yards  of  a  matting  2  feet  6  inches  broad, 
will  cover  a  floor  that  is  27  feet  long,  and  20  broad? 

Ans.  72  yards. 

11.  If  a  person's  annual  income  be  520  dollars,  what 
is  that  per  week?  Ans.  10  dollars, 

12.  If  a  pasture   be  sufficient  for  3000  horses  for  18 
days,  how  Jong  will  it  be  sufficient  for  2000? 

H.          H.  D.      D. 

As  2000   :  3000   :    :   18   :  27  Ans. 

13.  What  must  be  the  length  of  a  piece  of  land  13| 
rods  in  breadth,  to  contain  one  acre? 

Ans.  11  rods,  4yds.  2ft.  Oifin. 

*  The  sum  in  the  third  example  is  read  thus:— As  450  is  to  8,  so  is  1350  to  the 
answer    This  is  the  manner  of  reading  all  sums  when  stated  in  the  Rule  of  Three. 


108  SINGLE   RULE   OF   THREE. 

14.  If  8  men  can  build  a  tower  in  12  days,  in  what 
time  can  12  build  it? 

M.     M.        D.     D. 
As  12    :  8   :    :   12   :  8  Answer. 

15.  If  a  piece  of  land  be  5  rods  in  width,  what  must 
be  its  breadth  to  make  an  acre? 

R.      R.          R.     PC. 
As  5   :  160   :   :  1    :  32  Answer. 

16.  How  much  carpeting  that  is  IJ  yards  in  breadth, 
will  cover  a  floor  7\  yards  in  length,  and  5  yards  in 
breadth? 

By  Decimal  Fractions, 
yds.  yds.  yds. 

As  1.5   :  5   :   :  7.5   :  25  Answer. 

17.  What  will  one  quart  of  wine  cost  at  the  rate  of  12 
dollars  for  16  gallons? 

gals.     qts.     qt.  D.          cts. 

As  16  or  64   :   1   •.   :  12.00   :  18J  Answer. 

18.  If  10  pieces  of  cloth,  each  piece  containing  42 
yards,  cost  531  dollars  30  cents,  what  does  it  cost  per 
yard?  Ans.  $  1.261. 

19.  If  a  hogshead  of  brandy  cost  78  dollars  75  cents, 
what  must  be  given  for  5  gallons  at  the  same  rate? 

Ans.  $6.25. 

20.  If  a  staff  4  feet  in  length,  cast  a  shade  on  level 
ground,  8  feet  in  length,  what  is  the  height  of  a  tower 
whose  shade,  at  the  same  time,  measures  200  feet? 

ft.      ft.         ft.      ft. 
As  8    :  200   :    :  4   :   100  Answer. 

21.  I  lent  my  friend  350  dollars  for  5  months,  he  pro- 
mising to  do  me  the  same  favour;  but  when  requested, 
he  could  spare  only  125  dollars.     How  long  ought  1  to 
keep  it  to  balance  the  favour? 

D.       D.          M.    M. 
As  125   :  350   :    :  5   :   14  Answer. 

22.  If  7  oxen  be  worth  10  cows,  how  many  cows  will 
-21  oxen  be  worth? 

Ox.    Ox.         C.      C.      . 
As  7   :  21    :   :   10   :  30  Answer 


SINGLE    RULE    OF    THREE.  109 

23.  If  48  men  can  build  a  fortification  in  24  days,  how 
many  men  can  do  the  same  in  192  days? 

D.       D.         M.     M. 
As  192   :  24    :    :  48    :  6  Answer. 

24.  A  certain  piece  of  work  was  done  by  120  men  in  8 
months,  how  many  men  will  it  take  to  do  another  piece 
of  work  of  the  same  magnitude  in  2  months?   Ans.  480. 

25.  A  merchant  failing  in  trade,  owes  29475  dollars: 
he  delivers  up  his  property  which  is  worth  21894  dol- 
lars 3  cents;  how  much  does  this  sum  pay  on  the  dollar 
towards  what  he  owes?  Ans.  74cts.  2m.-|~ 

26.  If  a  tax  of  30,000  dollars  be  laid  on  a  town  in 
which  the  ratable  property  is  estimated  at  9,000,000  dol- 
lars, what  will  be  the  tax  of  one  of  the  citizens  whose 
ratable  estate  is  reckoned  at  750  dollars. 

D.  D.  D.      D.  cts. 

As  9,000,000   :  30,000   :   :  750   :  2  .  50  Ans* 

27.  If  property  rated  at  $28,  pay  a  tax  of  $21,  how 
much  is  that  on  the  dollar? 

D.     D.         D.     cts. 
As  28   :  21    :   :  1    :  75  Answer. 

28.  How  far  are  the  inhabitants  on  the  equator  carried 
in  a  minute,  allowing  the  earth  to  make  one  revolution 
in  24  hours;  and  allowing  a  degree  to  contain  691  miles? 

The  earth  being  divided  into  360  degrees,  allowing 
69^  miles  to  a  degree,  makes  the  distance  round  it  to  be 
25020  miles; — the  number  of  minutes  in  24  hours  is 
1440:  then, 

min.     min.       miles,     miles,  fur. 
As  1440   :   1    :   :  25020   :   17  .   3  Answer. 

29.  There  is  a  cistern  having  4  spouts;  the  first  will 
empty  it  in  15  minutes,  the  second  in  30  minutes,  the 

*  In  making  taxes  in  a  due  proportion,  according  to  the  value  of  each  man's  ra- 
table estate,  proceed  in  the  following  manner.  Make  the  amount  of  ratable  pro 
perty  the  first  term;  make  the  sum  to  be  raised  the  second  term;  and  one  dollar  the 
third  term;  and  the  number  arising  from  this  operation  will  be  the  amount  to  be 
raised  on  the  dollar.  From  this,  make  a  tax  table  from  one  dollar  to  30,  or  any 
amount  necessary.  In  the  same  manner  find  what  is  to  be  paid  on  a  cent  of  rata- 
ble estate;  and  from  this,  mako  a  table  from  1  to  99  cents;  then,  from  these  tableg, 
take  each  man's  tax.  Thus,  if  the  tax  were  75  cents  on  the  dollar,  and  you  would 
know  what  a  portion  of  property  pays,  that  is  rated  at  $28.80,  the  tables  will  show 
the  amount  to  be  $21,  for  the  dollars,  and  60  cts.,  for  the  cents.  In  estimating  pro- 
perty for  making  taxes,  it  is  customary  to  rate  it  much  lower  tha»  its  rea!  value. 

10 


i'10  SINGLE    RULE    OF    THREE. 

third  in  45  minutes,  and  the  fourth  in  60  minutes:  in 
what  time  would  the  cistern  be  emptied,  if  they  were 
all  running  together? 

As   15    :   1    :       90    :  6 


30  :  1 
45  ;  1 
60  :  1 


90  :  3 
90  :  2 
90  :  li 

12J 


cisterns,  cist.     min.  min.sec. 

Then,  decimally,  as  12.5  :  1  :  :  90  :  7  .  12  Ans. 
30.  If  a  ship's  company  of  15  persons  have  a  quanti- 
ty of  bread,  sufficient  to  afford  to  each  one  8  ounces  per 
day,  during  a  voyage  at  sea,  what  ought  to  be  their  al- 
lowance, under  the  same  circumstances,  if  5  persons  be 
added  to  their  number.  Ans.  6  ounces. 

Note. — As  the  Rule  of  Three  in  Vulgar  and  Decimal 
Fractions  require  the  same  statements  as  in  whole  num- 
bers, and  is  performed  by  multiplication  and  division 
after  the  sanae  manner  of  other  sums  in  the  Rule  of 
Three,  it  is  deemed  unnecessary  to  give  any  examples. 
When  the  pupil  understands  Fractions  and  the  Rule  of 
Three,  he  will  find  no  difficulty  with  the  Rule  of  Three 
in  Fractions. 

Ct-  1.  What  is  the  Rule  of  Three  sometimes  called? 

2.  What  does  it  teach? 

3.  Which  of  the  terms  must  be  set  in  the  third  place? 

4.  How  do  you  ascertain  which  ought  to  be  the  first 

term,  and* which  the  second? 

5.  If  the  third  term  consist  of  different  denomina- 

tions, what  do  you  do  with  them? 

6.  What  do  you  do  if  the  first  and  second  terms  are 

of  different  denominations? 

7.  After  stating  the  sum,  and  reducing,  when  neces 

sary,  the  terms  to  similar  denominations,  how 
do  you  proceed  to  do  the  sum? 
3.  How  are  sums  in  the  Single  Rule  of  Three  proved  •> 


DOUBLE  RULE  OF  THREE*  1H 

DOUBLE  RULE  OF  THREE. 

The  Double  Rule  of  Three  is  that  in  which  five  or 
more  terms  are  given  to  find  another  term  sought. 
RULE. 

Set  the  term  which  is  of  the  same  denomination  as 
the  term  sought,  in  the  third  place;  then  consider  each 
pair  of  similar  terms  separately,  and  this  third  one,  as 
making  the  terms  of  a  statement  in  the  Single  Rule  of 
Three,  setting  the  similar  terms  in  the  first  or  second  pla- 
ces, according  to  the  rule  of  the  Single  Rule  of  Three. 
After  stating  the  question  in  this  manner,  and  reducing, 
if  necessary,  the  similar  terms  to  similar  denominations, 
then  multiply  the  terms  in  the  second  and  third  places 
together  for  a  dividend,  and  the  terms  in  the  first  place 
together  for  a  divisor — the  quotient,  after  dividing,  will 
be  the  term  sought. 

Sums  in  this  rule  may  also  be  done  by  two  or  more 
statements  in  the  Single  Rule  of  Three. 
PROOF. 

By  inverting  the  statement,  or,  more  easily,  by  two 
statements  in  the  Single  Rule  of  Three. 
EXAMPLES. 

1.  If  8  men,  in  16  days,  can  earn  96  dollars,  how  rritich 
can  12  men  earn  in  26  days? 

men      8      :      12      :      :      )   ^ 
days   16      :     26      :      :      \   ™ 

128          312 

96X 


1872 
2808 

D. 

128)29952(234  Anfewer. 
256 

435 
384 

512 
512 


112  DOUBLE  RULE  OF  THREE* 

F  $100  gain  $6  in  12  months  w 
Miths? 

months     12    :  9    :   :    \  $400 


2.  If  $100  gain  $6  in  12  months  what  will  $400  em 
in  9  months? 


1200    54 

400  X 


12|00)216[00 


Answer. 

3.  If  16  men  can  dig  a  trench  54  yards  in  length  in 
o  days,  how  many   men  will  be  necessary  to  complete 
one,  135  yards  in  length  in  8  days? 

By  two  statements  in  the  Single  Rule  of  Three* 

yds.     yds.        men.    men. 
As  54   :  '135    :    :   16    :  40. 

days.  ds.       men.  men. 
Then,  as  8   :  6   :   :  40   :  30  Answer. 

4.  If  $100  in  one  year  gain  $5  interest,  what  will  be 
*he  interest  of  $750  for  7  years?  Ans.  $262.50. 

5.  If  9  persons  expend  $120  in  8  months,  how  much 
will  24  persons  spend  in  16  months  at  the  same  rate? 

AHS.  $640. 

6.  If  54  dollars  be  the  wages  of  8  men  for  14  days, 
what  must  be  the  wages  of  28  men  for  20  days  at  the 
same  rate?  Ans.  $270, 

7.  If  a  horse   travel   130  miles  in  3  days,  when  the 
days  are  12  hours  in  length,  in  how  many  days  of  10 
hours  each  can  he  travel  360  miles?         Ans.  9||  days. 

8.  If  60  bushels  of  corn  can  serve  7  horses  28  days3 
how  many  days  will  47  bushels  serve  6  horses? 

Ans.  51?8j  days. 

9.  If  a  barrel  of  beer  serve  7  persons  for  12  days, 
how  many  barrels  will  be  sufficient  for  14  persons  for  a 
year,  or  365  days?  Ans.  60 J  barrels. 

10.  If  8  men  spend  32  dollars  in  13  weeks,  what  will 
24  men  spend  in  52  weeks? 

*  When  a  sum  in  the  Double  Rule  of  Throe  appears  difficult  to  be  stated  for  owe 
operation,  it  may  uiways  be  done  with  ease  by  two  statements  in  the  Single  Rwle 
•>f  Tljp'e,  us  in  the  above  example. 


PRACTICE.  113 

By  two  statements  in  the  Single  Rule  of  Tliree. 

men.  men.       D.      D. 
As  8   :  24   :   :  32    :  96 

weeks,  weeks.      D.       D. 
Then,  as  13   :  52   :   :  96   :  384  Answer. 

Q.  1.  How  many  terms  are  generally  given  in  the  Dou- 
ble Rule  of  Three? 

2.  Which  of  the  terms  must  be  set  in  the  third  place? 

3.  How  do  you  ascertain  which  of  the  other  terms 

should  be  placed  in  the  first,  and  which  in  the 
secend  place? 

4.  Which  of  the  terms  do  you  multiply  together  for 

a  dividend? 

5.  How  do  you  form  a  divisor? 

6.  How  do  you  proceed  when  the  terms  consist  of 

different  denominations? 

7.  By  what  other  method  may  sums  be  done  in  the 

Double  Rule  of  Three,  besides  the  one  first 
given? 

8.  How  is  a  sum  in  the  Double  Rule  of  Three  proved? 


PRACTICE. 

Practice  is  a  short  method  of  doing  all  sums  in  the 
Single  Rule  of  Three,  that  have  one  for  their  first  term, 
and  is  of  great  use  among  merchants. 

It  may  be  proved  by  Compound  Multiplication,  or  by 
the  Single  Rule  of  Three. 

Questions  to  prepare  the  learner  for  this  rule. 

1.  What  will  50  yards  of  tape  cost  at  £  of  a  cent  per 

yard? 

2.  What  will  40  pounds  of  beef  come  to  at  }  of  a 

eent  per  pound? 

3.  What  will  100  figs  come  to  at  f  of  a  cent  a  piece? 

4.  How  many  pence  will  40  peaches  come  to  at  one 

farthing  a  piece? 

5.  How  many  shillings  and  pence  will  50  peaches  come 

to  at  one  farthing  a  piece? 
10* 


114 


PRACTICE. 


PRACTICE  TABLE,  OR  TABLE  OF  ALIQUOT 


cts. 

dols.              s. 

d. 

£-               d. 

^. 

50 

=  i 

10 

0 

2" 

1      _.- 

JL 

25 

o 
*** 

6 

8 

o 

li 

V 

20 

5" 

P0 

'5 

0 

I 

2 

121 

•1 

Q 

4 

0 

A 

P 

3 

1 

4 

4 

rV 

P 

3 

4 

6" 

.     O 
Q 

4 

5 

iV 

2 

6 

-I 

D 

6 

JL 

4 

AJ 

1 

8 

TV 

P* 

qrs. 

/^. 

m. 

cis. 

1 

0 

A  J 

2  or 

56 

5 

i  )    2, 

0r. 

rf. 

1 

28 

2 

i       «» 

2 

*  I  d 

16 

1 

rV)  a 

1 

i  1 

14 

8 

7 

cwt. 


TV 


CASE   I. 


«  price  o/"  owe  yard,  pound,  fyc.  is  in  farthings, 

RULE. 

Divide  by  the  aliquot  parts  of  a  penny,  and  the  an- 
swer will  be  in  pence,  which  reduce  to  shillings,  pounds, 
&c. 

EXAMPLES. 

1.  What  is  the  value  of          2.  What  is  the  value  of 
380  at  one  farthing  each?        744  at  3  farthings  each? 
380  2  1  i  1  744 


12)95  pence.  1 

7s.  lid.  Ans. 

3.  What  is  the  value  of 
460  at  2  farthings  each? 
460 

12)230  pence. 

19s.  2d.  Answer 


372 

186 


12)558 
20)46—6 

£2  6s.  6d.  Am* 


>ftACTlCE. 


4.  What  is  the  worth  of  298  at  id.?        Ans.  6s.  2-£d. 

5.  What  is  the  worth  of  586  at  id.?    Ans.  £1  4s.  5d, 

6.  What  is  the  worth  of  964  at  Jd.?    Ans.  £3  Os.  3d, 

CASE    II. 

When  the  price  is  any  number  of  pence  less  than  12. 

RULE. 

Divide  by  the  aliquot  parts  of  a  shilling,  and  the  an- 
swer will  be  in  shillings,  which  may  be  reduced  to 
pounds. 

EXAMPLES. 


1 


i. 

TV 


II. 


672  at  Id. 
20)56 

£2  16s.  Ans. 


444  at 


20)74 


£3  14s.  Ans. 


3.  What  is  the  value  of  237  at  3d.? 

4.  What  is  the  value  of  594  at  4d,? 

5.  What  is  the  value  of  868  at  6d.? 

6.  What  is  the  value  of  988  at  5d.? 


£  s.  d. 

Ans.  2  19  3 
Ans.  9  18  0 
Ans.  21  14  0 
Ans.  20  1 1  8 


7,  What  is  the  value  of  1049  at  8d.?      Ans.  34  19  4 

8.  What  is  the  value  of  1294  at  10d.?   Ans.  53  18  4 


4d. 
Id. 


988 


at  5d. 


20)411 


£20  11s.  8d.  Answer. 

Note. — In  this  last  example,  as  5d  is  not  an  aliquot 
part  of  a  shilling,  I  take  4  pence,  which  is  one  third  of 
a  shilling,  and  after  dividing  by  that,  I  take  one  penny, 
which  is  one  fourth  of  four,  and  divide  the  first  product 
by  it.  Then,  adding  them  together,  and  reducing  them 
to  pounds,  &c,  I  obtain  the  answer, 


116 


PRACTICE. 


CASE    III. 
When  the  price  in  pence  exceeds  the  number  of  12. 

RULE. 

Consider  the  number  given  in  the  sum  as  containing 
so  many  shillings.  Then  divide  by  such  aliquot  parts 
as  may  be  formed  by  the  pence  over  a  shilling,  adding 
the  product  to  the  sum.  The  answer  will  be  in  shillings, 

EXAMPLES. 


600      at  13|d. 
75 


20 


675 


£33  ISs.  Answer. 
Note. — In  this  example,  I  consider  the  sum  as  600 
shillings.  Then,  as  the  given  price  is  lid.  over  a  shil- 
ling, which  makes  |  of  a  shilling,  I  divide  the  sum  by  8, 
and  add  the  quotient  to  the  given  sum  j  which  makes  675 
shillings,  or  £33  15s. 

£    s.    d. 

2.  What  is  the  worth  of  450  at  14d.?  Ans.  26     5     0 

3.  What  is  the  worth  of  570  at  16d.?  Ans.  38     0    0 

CASE    IV. 

When  the  price  is  any  number  of  shillings  under  20. 
RULE. 

Divide  by  the  aliquot  parts  of  a  pound,  and  the  an- 
swer will  be  in  pounds.  Or,  consider  the  sum  as  being 
so  many  shillings,  then  multiply  the  sum  by  the  number 
of  shillings  in  the  price.  The  product  will  be  the  an- 
swer in  shillings;  which  reduce  to  pounds. 

EXAMPLES. 


5s. 


1296  at  5s. 
£324  Ans. 


n. 

723     at  12s. 
12X 


20)8676 


£433  16s.  Answer. 


PRACTICE* 


117 


The  second  example  is  done  by  the  second  method, 
which  is  thought  by  many  to  be  the  easier  way. 

£ 


3.  What  is  the  value  of  1128  at  3s.?      Ans.    169 

4.  What  is  the  value  of    889  at  4s.?      Ans.    177 

5.  What  is  the  value  of  1616  at  9s.?      Ans.    727 


s. 

4 
16 

4 


S.  What  is  the  value  of  2868  at  18s.?    Ans.  2581     4 
CASE    V. 

When  the  price  is  in  pounds,  shillings  and  pence. 

RULE. 

Multiply  the  gum  or  quantity  by  the  number  of  pounds 
in  the  price,  then  divide  by  the  aliquot  parts  of  shillings 
and  pence,  and  add  the  quotients  to  the  product — their 
sum  will  be  the  answer. 

EXAMPLES. 

I.  II. 


10 


448  at  £41  Os.  6d. 
4 


4 

i 

5 

5678  at  £7 

4s.  9d, 

7 

39746 

6 

1_ 

1135  12 

3 

JL 
2 

141   19 

70  19 

6 

1792 
224 
11      4 

£2027     4s.  Ans. 

£41094     10s.  6d. 

Note. — In  the  second  example,  after  multiplying  the 
sum  by  the  number  of  pounds,  as  4s.  is  i  of  a  pound,  I 
divide  by  5,  which  gives  1135. pounds  in  the  quotient; 
and  leaving  a  remainder  of  3  pounds,  which  reduced  to 
shillings  and  divided  by  5  give  12s.  Then,  as  6  and  3 
make  9,  the  number  of  pence,  and  as  6d.  is  |  of  4s.,  I 
divide  the  quotient  by  8,  which  gives  141  pounds  with 
a  remainder  of  7;  this  being  reduced  to  shillings,  and 
the  12  shillings  above  added  to  it  make  152,  which  di- 
vided still  by  the  8  give  19  shillings.  And  as  3  is  \  of 
6,  or  its  aliquot  part,  I  divide  the  last  quotient  by  2. — 
This  Drives  70  pounds  and  a  remainder  of  one,  which  is 
20  shillings;  and  adding  it  with  19  shillings  above,  the 
amount  is  39  shillings.  This  divided  by  the  2  gives  19 


118 


PRACTICE. 


shillings  and  a  remainder  of  1  shilling,  or  12  pen<ff. ' 
which,  divided  still  by  the  2,  makes  6d.  And  thus  the 
answer  is  obtained. 

3.  What  is  the  amount  of  288  at  £5  10s.  4d.? 

Ans.  £1588  16s, 

4.  What  is  the  amount  of  642  at  £9  4s.  6d.? 

Ans.  £5922  9s. 

5.  What  is  the  amount  of  734  at  £12  2s.  8d.? 

Ans.  £8905  17s.  4cl 
CASE    VI. 

IVken  the  quantity  consists  of  different  denominations,  and 
the  price  is  in  pounds?  shillings^  <$'C. 

RULE. 

Multiply  the  price  of  the  highest  denomination  given, 
by  the  whole  of  the  highest  denomination,  then  divide 
by  aliquot  parts  of  each  of  the  lower  denominations  in 
the  sum.  Add  the  results  together,  and  their  sum  will 
be  the  answer. 

EXAMPLES. 


£ 


at      4 

6 

2  per  cwt 

i 

S 

12 

18 

6 

1 

2 

3 

1 

10 

9* 

3cwt.  2qrs.  14  Ibs. 
2  qre.  are  £  of  a  cwt. 


14  Ibs.  are  £  of  2  qrs. 


£15  12s.  4£d.  Ans. 

2.  4  cwt.  3  qrs.  12  Ibs.  at  £8  4s.  4d.  per  cwt. 

Ans.  £39  18s.  2d. 

3.  5  cwt.  3  qrs.  4  Ibs.  at  £9  6s.  8d.  per  cwt. 

Ans.  £51  18s.  lOJd, 

4.  7  cwt.  0  qr.  14  Ibs.  at  £2  3s.  4d.  per  cwt. 

Ans.  £15  8s.  9d. 

5.  8  cwt.  3  qrs.  24  Ibs.  at  £1  2s.  3d.  per  cwt. 

Ans.  £9  19s.  5]d, 

6.  9  cwt.  1  qr.  18  Ibs.  at  £3  10s.  lOd.  per  cwt. 

Ans.  £33  6s.  7d, 

7.  10  cwt.  2  qrs.  10  Ibs.  at  £4  4s.  6d.  per  cwt. 

Ans.  £44  14s,  9]d 


3PBACT1CE. 


119 


FEDERAL   MONEY. 
CASE    I. 

When  the  price  is  -j,  1,  or  |  of  a  cent. 

RULE. 

Divide  the  sum  by  the  even  parts  of  a  cent,  and  the 
answer  will  be  cents. 

EXAMPLES. 

1.  What  is  the  worth  of  452  Ibs.  at  \  cent  per  lb.? 


i         - 

L     452  at  £  cent. 

226  cents, 

or  $2.26.  Answer. 

II. 

in. 

1 

4 

2468  at  i  cent  each?         1 

JL 

2 

987654  at  f  ct.? 

617  cents. 

or  $6.17  Ans. 

| 

\ 

493827 

2469131 

$7407.401  Ans, 
CASE    II. 
When  the  price  is  in  cents. 

RULE. 

Divide  the  sum  by  the  aliquot  parts  of  a  dollar,  and 
the  answer  will  be  in  dollars. 

EXAMPLES. 

1.  What  is  the  worth  of  2345  yards  at  20  cents  per 
yard? 


20 


2345 


Answer. 


25     i     348  at  25  cents. 

$87— Ans. 
CASE    III. 

When  the  price  is  dollars  and  cents. 

RULE. 

Multiply  the  quantity  by  the  dollars,  then  work  for 
the  cents  as  in  the  last  case,  add  the  products  together 
for  the  answer. 

EXAMPLES. 

-  1.  What  is  the  worth  of  5220  at  three  dollars  and 
fwenty  cents  each? 


120 

PRACTICE. 

20 

i 

52. 

20                                 II. 

3                  50 

i1 

8.48  at 

$5.50. 

— 

5 

" 

156. 

60 

10. 

44 

42 

.40 

:  , 

.,  — 

4 

.24 

$  167.04 

— 

— 

$46.64  Answer 
CASE   IV. 

When  the  price  is  no  aliquot  part  of  a  dollar. 

RULE. 

Divide  by  two  or  more  numbers,  whose  sum  will  make 
the  number  wanted. 

EXAMPLES. 

1.  W^hat  will  9754  Ibs.  cost  at  30  cents  per  lb.? 
"  9754 


>  cts. 
10  cts.  i  of  20 


1950.80 
975 


$2925.80  Answer. 
2.  What  will  642  Ibs.  cost  at  60  cents  per  lb.? 

Ans.  $385.20. 
CASE   V. 

When  there  are  several  denominations  in  the  quantity ,  and 
the  price  is  dollars  and  cents. 

RULE. 

Multiply  the  dollars  in  the  price  by  the  number  of 
the  highest  denomination  in  the  quantity;  work  for  the 
cents  by  the  rules  in  the  preceding  cases;  for  the  parts 
in  the  quantity,  take  aliquot  parts  of  each  lower  denomi- 
nation, and  add  the  products  together. 

EXAMPLES. 

1.  What  is  the  value  of  20cwt.  3  qrs.  14  Ibs.  at  12 
dollars  and  25  cents  per  cwt.? 
25  cts.  =  I  i  I     20  cwt. 

I     12  dollars. 

240  price  at  12  dollars. 
5  price  at  25  cents. 

$245  price  of  20  cwt,  at  $12.25. 


FELLOWSHIP.  121 


To  obtain  the  parts  in  the  quantity. 

cwt. 
2  qrs.  =     i     12.25 


1  qr.  = 
14  Ibs.  = 


6.121  price  of  2  qrs. 
3.061  price  of  1  qr. 
1.53     price  of  14  Ibs. 

10.71|  price  of  3  qrs.  14  Ibs. 
245.00     price  of  20  cwt. 

$255.7 If  Answer. 

2.  What  is  the  worth  of  64  cwt.  2  qrs.  141bg.  at  16 
dollars  and  20  cents  per  cwt.?  Ans.  $1046.921  . 

Q.  1.  What  is  "practice? 

2.  Wherein  is  it  particularly  useful? 

3.  Repeat  the  table  of  aliquot  parts. 

4.  How  many  cases  are  there  in  pounds,  shillings 

&c.? 

5.  Repeat  the  rule  of  each  different  case. 

6.  What  number  of  cases  do  you  find  in  Federal 

Money? 

7.  Repeat  the  rule  of  each  case. 

8.  How  are  sums  in  Practice  proved? 


FELLOWSHIP. 

Fellowship  is  an  easy  rule  by  which  merchants  or 
other  persons  in  company,  are  enabled  to  make  a  just 
division  of  the  gain  or  loss  in  proportion  to  each  per- 
son's share.  Sums  in  Fellowship  are  generally  done  by 
the  Rule  of  Three. 

CASE   T. 

When  the  several  shares  are  considered  without  regard  to 
time. 
RULE. 

As  the  sum  of  all  the  stock  is  to  each  person's  parti- 
cular share  of  the  stock,  so  is  the  sum  of  all  the  gain  01 
loss,  to  the  gain  or  loss  of  each  person. 
11 


13-2  FELLOWSHIP. 

PROOF. 

Add  together  all  the  shares  of  gain  or  loss,  and  if  it 
be  right,  the  sum  will  be  equal  to  the  whole  gain  or  loss. 

EXAMPLES. 

1.  A  and  B  purchase  certain  goods  amounting  to  $580. 
of  which  A  pays  $350  and  B  $230.     They  gain  262;— 
what  is  each  man's  share  of  the  gain? 

A  $350 
B  $230 

A's  share,     gain.       $  cts. 

As    580   :  350   :    :  262    :   158.  lOf £  A?s  gain. 

B's  share,     gain. 
Then,  as  580   :  230   :    :  262    :   103.89f|  B's  gain. 

2.  A,  B  and  C  formed  a  company.     A  put  in  $40,  B 
60  and  C  80.     They  gained  $72 : — what  was  each  man's 
share?  Ans.  A  gained  $16,  B  24  and  C  32. 

3.  A,  B  and  C  lose  a  quantity  of  property   worth 
$2400;  of  which  A  owned  |,  B  -J-  and  the  remainder  to 
C;  what  does  each  lose? 

Ans.  A  loses  $600,  B  800  and  C  1000. 

4.  A  and  B  have  gained  $800,  of  which  A  was  to  re- 
ceive 10  per  cent,  more  than  B;  what  did  each  receive? 

Ans.  A  received  $440  and  B  360. 

5.  A  and  B  purchase  goods  worth  $80,  of  which  A 
pays  30  and  B  50.     They  gain  $20; — what  is  the  gain 
of  each?  Ans.  A's  gain  is  $7.50  and  B's  12.50. 

6.  Four  men  formed  a  capital  of  $3200.  They  gained 
in  a  certain  time  $6560.     A's  stock  was  $560,  B's  1040, 
C's  1200  and  D's  400.     What  did  each  gain? 

Ans.  A's  gain  was  $1 148,  B's  2132,  C's  2460  and  D's  820. 

CASE    II. 

When  the  different  stocks  in  company  are  considered  in  re- 
lation to  time. 

RULE. 

Multiply  each  man's  stock  by  the  time  it  has  been  a 
part  of  the  whole  stock:  then,  as  the  sum  of  the  pro- 
ducts is  to  either  single  product,  so  is  the  whole  sum  of 
gain  or  loss  to  the  gain  or  lass  of  each  man. 


TARE  AND  TRET.  123 

EXAMPLES. 

1.  A,  B  and  C  hold  a  pasture  in  common,  for  which 
they  pay  $40  per  annum.  A  put  in  9  cows  for  five  weeks; 
B,  12  cows  for  7  weeks;  and  C,  8  cows  for  16  weeks.*-- 
What  must  each  man  pay  of  the  rent? 

9X   5  =    45 

12X   7  =     84 

8X16=  128 

Dolls.     Dolls. 

As  257  :  45  :  :  40  :  7^  A'a  part. 
As  257  :  84  :  :  40  :  13^  B's  part. 
As  257  :  128  :  :  40  :  19ffX  C's  part. 

2.  A  with  a  capital  of  £1000,  entered  into  business 
<jn  the  first  of  January.     On  the  first  of  March  follow- 
ing he  took  in  B  as  a  partner,  who  brought  with  him  a 
capital  of   £1500;    and  three  months  after  they  are 
joined  by  C  with  a  capital  of  £2800.     At  the  end  of  the 
year,  they  find   they  have  gained  £1776  10s.     How 
must  it  be  divided  among  them? 

Ans.  A's  part  will  be  £457    9s.    4£d. 
B's  part  will  be  £571  16s.     8£d. 
C's  part  will  be  £747    3s.  Hid. 
Q.  1.  What  is  Fellowship? 

2.  By  what  rule  are  sums  in  Fellowship  usually  done? 

3.  How  do  you  proceed  when  the  shares  are  consi- 

dered without  regard  to  time? 

4.  How  do  you  proceed  when  the  shares  are  consi- 

dered in  relation  to  time? 

5.  How  are  sums  in  Fellowship  proved? 


TARE  AND  TRET. 

Tare  and  Tret  are  certain  allowances  made  by  mer- 
chants in  selling  their  goods  by  weight. 

Tare  is  an  allowance  made  for  the  weight  of  the  bar- 
rel, bag,  &c.,  that  contains  the  article  or  commodity 
bought. 

Tret  is  an  allowance  of  4  Ibs.  in  every  104  Ibs.  f&f- 
waste,  dust,  &c. 


124  TARE    AND    TRET* 

Gross  weight  is  the  weight  of  the  goods,  together 
with  the  barrel,  box,  or  whatever  contains  them.  When 
the  tare  is  deducted  from  the  gross,  what  remains  is 
called  suttle. 

Neat  weight  is  the  weight  of  articles  after  all  allow- 
ances are  deducted. 

CASE    I. 

When  the  fare  is  so  much  per  hhd.  on  any  given  quantity, 

RULE. 

Subtract  the  tare  from  the  quantity — the  remainder 
will  be  the  neat  weight. 

EXAMPLE. 

In  6  hhd.  of  sugar,  each  weighing  9  cwt.  2  qrs.  10  Ibs, 
gross,  tare  25 Ibs.  per  hhd.  how  much  neat  weight? 

cwt.  qr.      Ib.  cwt.  qrs.     Ibs. 

25X6=1        1       10  tare.  9       2       10 

6X 


57       2         4  gross 
1       1       10  tare. 


56       0       22  An«, 
CASE    IT. 

When  the  tare  is  at  so  much  per  cwt. 

RULE. 

Divide  the  gross  weight  by  the  aliquot  parts  of  a  cwt, 
then  subtract  the  quotient  from  the  gross,  and  the  re- 
mainder will  be  the  neat  weight. 

EXAMPLES. 

1.  In  129  cwt.  3  qrs.  161bs.  gross,  tare  14  Ibs.  per 
cwt.  what  neat  weight? 


14  Ibs. 


129     3     16    gross. 
16     0     261 


113     2     17-J  Answer. 
2.  In  97  cwt.   1  qr.  7  Ibs.  gross,  tare  20  Ibs.  per  cwt.. 
what  neat  weight? 


TARE    AND    TRET.  125 


16  Ibs. 
4  Ibs. 

Subti 

-act 

97 

1 

7 

gross. 
J  Add. 
[-  tare. 

13 

3 

3 
1 

25 

17 

1 

14, 

79 

3 

20|  Answe 

./Vote. — When  the  tare  per  cwt.  is  not  an  aliquot  part, 
the  tare  may  be  found  by  the  Rule  of  Three,  thus — As 
112  is  to  the  number  of  pounds  gross,  so  is  the  rate  per 
cwt.,  to  the  tare  required. 

3.  What  is  the  neat  weight  of  38  cwt.  0  qr.  4  Ibs.  tare 
at  1 1  Ibs.  per  cwt. 
cwt.  qr.  Ibs. 
38     0     4  =  4260  pounds. 

Ibs.          Ibs.  Ibs. 

Then,  as  112   :  4260   :   :  11    :  4 18T4TV  Answer. 
4260 
418/ft 

cwt.  qrs.  Ibs. 

3841T6T83  neat  =  34     1     5TfiT^  Answer, 

CASE    III. 

When  tare  and  tret  are  allowed. 

RULE. 

Find  the  tare  according  to  the  preceding  rules,  sub- 
tract it  from  the  gross,  and  the  remainder  will  be  suttle; 
then  divide  the  suttle  by  26,  and  the  product  will  be  the 
tret,  which,  subtract  from  the  suttle — the  remainder  will 
be  the  neat. 

Note. — As  4  pounds  on  the  104  Ibs.  is  the  customary 
allowance  for  tret,  we  divide  by  26,  because  4  is  -^  of 
104. 

EXAMPLES. 

1.  In  247  cwt.  2  qrs.  15  Ibs.  gross,  tare  28  Ibs.  per  cwt, 
and  tret  4  Ibs.  for  every  104  Ibs.  how  much  neat? 


1 26 


28  Ibs.  = 


4lbs.=  |  aVof  104 


247     2     15 
61     3     17     12  tare  subtract. 


185     2     25       4 
7     0     16       0 


Ans.  178     2       9       4  neat. 

2.  In  9cwt.  1  qr.  lOlbs.  gross,  tare  28  Ibs.  per  cvt 
and  tret  4  Ibs.  for  every  104  Ibs.  how  much  neat? 

Ans.  6cwt.  2  qrs.  26ilbsL 

3.  A  merchant  purchased  4  hhds.  of  tobacco,  weigh- 
ing  as  follows: — The  first  5  cwt.  1  qr.  12  Ibs.  gross,  tare 
65  Ibs.  per  hhd.;  the  2d.  3  cwt.  Oqr.  19  Ibs.  gross,  tare 
75  Ibs.;  the  3d.  6  cwt.  3  qrs.  gross,  tare  49  Ibs.;  the  4th 
4  cwt.  2  qrs.  9  Ibs.  gross,  tare  35  Ibs.  and  allowing  tret 
to  each  at  the  rate  of  4  Ibs.  for  every  104  Ibs.     What 
was  the  neat  weight  of  the  whole? 

Ans.  17  cwt.  0  qr.  19  Ibs.  2  oz: 

Q.  1.  What  do  you  understand  by  Tare  and  Tret  2 

2.  What  is  tare? 

3.  What  is  tret? 

4.  What  is  gross  weight? 

5.  What  is  neat  weight? 

6.  What  is  called  suttle? 


BARTER, 

Barter  is  the  exchange  of  one  commodity  for  another, 
and  leaches  merchants  to  proportion  their  quantities 
without  loss. 

Questions  in  Barter  are  solved  either  by  the  Rule  of 
Three,  or  by  Practice. 

When  a  quantity  of  one  commodity  is  to  be  bartered 
fcr  a  quantity  of  another,  first  find  the  value  in  money 
of  the  quantity  to  be  exchanged,  then  find  what  quanti- 
ty of  the  other  may  be  had  for  that  amount. 
EXAMPLES. 

1.  How  much  flour  at  g3  per  barrel  must  be  given  in 
exchange  for  100  hhds.  of  salt  worth  $4.80  cts.  per  hhd 


SIMPLE   INTEREST.  127 

$4.80 

100  hhd. 


$480.00  price  of  the  salt. 
dolls.      bar.  dolls.         bar. 

Then,  As     3      :     1      :     :     480     :     160  Answer. 

2.  Two  merchants  wish  to  make  an  exchange,  A  ha"fc 
30  cwt.  of  cheese,  at  £l.  3s.  6d.  per  cwt.  and  B  has  9 
pieces  of  cloth,  at  £3.  15s.  per  piece — which  must  re- 
ceive money,  and  how  much?  Ans.  B  must  pay  A  £1  10s. 

3.  A  has  150  bushels  of  wheat  at  $1.25  per  bushel, 
for  which  B  gives  65  bushels  of  barley,  worth  62^  cents 
per  bushel,  and  the  balance  in  oats  at  37  J  cts.  per  bushel ; 
what  quantity  of  oats  must  A  receive  from  B?  Ans.  39  If, 


SIMPLE  INTEREST. 

Interest  is  a  premium  paid  for  the  use  of  money.  In 
calculating  interest  on  money,  four  things  are  necessary 
to  be  considered,  viz.  the  principal,  the  time,  rate  per 
cent.,  and  amount. 

The  principal  is  the  money  lent  for  which  interest  is 
to  be  received. 

The  rate  per  cent,  per  annum  (by  the  year)  is  the  in- 
terest for  100  dollars  or  100  pounds  for  one  year. 

The  time  is  the  number  of  years,  months  or  days,  for 
which  interest  is  to  be  calculated. 

The  amount  is  the  sum  of  the  principal  and  interest^ 
when  added  together. 

Questions  to  prepare  the  learner  for  this  rule. 

1.  If  you  give  $6  for  the  use  of  $100  for  a  year;  how 
much  must  you  give  for  the  use  of  $50? 

2.  If  you  give  $6  for  the  use  of  $100  for  a  year;  how 
much  must  you  give  for  the  use  of  it  for  six  months? 

3.  How  much  for  3  months? — How  much  for  4  months? 
How  much  for  8  months? — How  much  for  9  months? 

4.  If  the  interest  of  $200  be  one  dollar  for  a  month; 
fcow  much  will  it  be  for  15  days?-^-How  much  for  10 
days? — How  much  for  20  days? 


128  SIMPLE    INTEREST. 

CASE    I. 

When  the  time  is  one  year,  and  the  rate  per  cent,  is  any 
number  of  dollars,  pounds,  #c. 

RULE. 

Multiply  the  principal  by  the  rate  per  cent ,  divide  the 
product  by  100,  and  the  quotient  will  be  the  interest  for 
one  year. 

EXAMPLES. 

1.  What  is  the  interest  of  328  dollars  for  one  year  at 
6  per  cent.? 

328  In  this  example,  as  cutting  off 

6  the  two  right  hand  figures  is  the 

same  as  dividing  by  100,  the  di- 
$19.|68cts.  Ans.         vision  is  omitted. 

2.  What  is  the  interest  of  $9876  for  one  year  at  6 
percent.?  6 

$592|56  cts.  Answer. 

When  the  sum  is  in  pounds,  if  there  be  a  remainder 
after  dividing,  or  after  cutting  off  the  two  right  hand 
figures,  the  remainder,  or  figures  cut  off  must  be  redu- 
ced to  shillings;  and  if  there  be  still  a  remainder  after 
dividing  the  shillings,  it  must  be  reduced  to  pence,  &c. 

3.  What  is  the  interest  of  £573  13s.  9id.  at  6  per 
cent,  per  annum? 

£573     13s.     9id.         ./Vote.— When  the  interest  is 
6          for  more  than  one  year,  mul- 
tiply the  interest  for  one  year 
£34|42       2     9  by  the  number  of  years.     To 

20  obtain  the  amount,  the  interest 

must  be  added  to  the  princi- 
8|42  pal. 

12 


5)13  £34.  8s.  5d.  Answer. 


SIMPLE    INTEREST,  129 

4.  What  is  the  interest  of  £40.  19s.  1  Id.  3qrs,  for  one 
vear,  at  6  per  cent,  per  annum? 

£40     19s.     lid.     3qrs. 
6 


2|45     19         10 
20 

9)19 
12 


2|38 
4 

1|52  remain.  Ans.  £2.  9s.  2d.  Iqr. 

5.  What  is  the  interest  of  87  dollars  for  one  year,  at 

6  per  cent,  per  annum?  Ans.  $5.22. 

6.  What  is  the  interest  of  143  dollars  for  one  year  at 

7  per  cent,  per  annum?  Ans.  $  10.01. 

When  the  rate  per  cent,  consists  of  a  whole  number 
and  a  fraction,  as  6£,  6|,  or  6J,  multiply  the  principal 
by  the  whole  number,  to  the  product  add  £,  or  i,  as  the 
case  may  be,  of  the  principal  and  then  divide  by  100,  or 
cut  off  the  two  right  hand  figures  as  before. 

7.  What  is  the  interest  of  228  dollars  for  one  year,  at 
6J-  per  cent,  per  annum? 


1368 
57 

$14|25cts.  Answer. 

When  the  principal  consists  of  dollars  and  cents,  mul- 
tiply by  the  rate  per  cent,  without  any  reference  to  the 
separating  point;  then  from  the  product  cut  off  the  first 
right  hand  figure  as  a  fraction  or  remainder,  the  next 
figure  will  be  mills,  the  two  next  cents,  and  the  other 
figures,  that  is,  those  on  the  left  of  the  cents,  will  be 
dollars, 


13®  SIMPLE    INTEREST. 

8.  What  is  the  interest  of  $98.79'  for  one  year,  at  6 
per  cent,  per  annum?  6    ' 

5|92|7|4  fraction. 

Ans.  $5  92c.  7m. 

9.  What  is  the  interest  of  432  dollars  73  cents  for  4 
years,  at  6  per  cent,  per  annum? 

$432.73 

6  rate  per  cent. 

259638 

4  number  of  years. 


103|85|5|2  frac.       Ans.  $103  85c.  5m, 

10.  What  is  the  interest  of  $8420.82  for  three  years, 
at  8  per  cent,  per  annum?  Ans.  $2020. 99c.  6m. 

11.  What  is  the  interest  and  amount  of  $7462.131  for 
four  years  at  7  per  cent,  per  annum? 

Ans.  Interest,  $2089.39c.  7m.     Am't.  $9551.53c.  2m, 

CASE   II. 

To  find  the  interest  when  the  given  time  is  months  or  days. 

RULE. 

Find  the  interest  for  one  year,  then  say — as  one  year 
is  to  the  given  time,  so  is  the  interest  of  the  sum  for  one 
year,  to  the  interest  for  the  time  required.  Or,  instead 
of  the  Rule  of  Three,  it  may  be  done  by  Practice, thus: 
For  the  number  of  months,  take  aliquot  parts  of  a  year; 
and  for  days,  the  aliquot  parts  of  30.* 

EXAMPLES. 

1.  What  is  the  interest  of  $98.50  for  9  months  and  18 
days,  at  6  per  cent,  per  annum? 
$98.50 
6 

$5.91|0|0  for  one  year. 
year.     mo.  days.  $  cts.          $  cts.  m. 

Then,  as   1      :     9     18      :     :     5     91      :     4  72  8  Ans. 

*  In  those  calculations,  a  j'ear  is  reckoned  at  360  (Jays,  and  a  month  at  30  days. 


SIMPLE    INTEREST.  131 

in  this  sum,  the  year  is  reduced  to  SCO  days,  the  9 
months  and  18  days  to  288  days,  and  the  third  term 
stands  as  591  cents. 

The  same  is  done  by  Practice,  thus — 

$98.50  ./Vote.— Interest  may 

6       be   calculated   in  the 
following  manner.viz : 


mo. 


6,  |  of  a  year. 

3.  -i  of  6  mo. 
15d.  iof  3  mo. 
3,  i  of  15  ds. 


5.91.010  When  rate  per  cent,  is 

9?  multiply  the  prin- 

2.95.5  cipal  by  Jof  the  given 

1.47.71  number  of  months; — 

24.61  when  it  is  8,  multiply 

4.9i  the  principal  by  |  of 
the  number  of  months: 


Ans.  $4.72.8  when  the  rate  is  6, 
multiply  by  1  the  number  of  months;  when  it  is  4,  multi- 
ply by  i;  when  it  is  3,  by  i;  and  when  the  rate  is  2, 
multiply  by  i — the  product  in  any  of  those  cases  will 
show  the  answer. 

2.  What  is  the  interest  of  $120.60  for  one  year  and 
three  months,  at  6  per  cent,  per  annum  ?  Ans.  $9. 04c.  5m. 

3.  What  is  the  interest  on  $187.061  for  10  months,  at 
6  per  cent,  per  annum?  Ans.  $9.35c.  3m. 

4.  What  is  the  interest  and  amount  of  640  dollars  for 
4  years  and  7  months,  at  5  per  cent,  per  annum? 

Ans.  g  146.66|  interest.     Am't.  $786.66|. 

5.  What  is  the  interest  of  $300  for  4  years,  4  months, 
and  20  days,  at  81  per  cent,  per  annum?  Ans.  $1 11.91|. 

6.  What  is  the~interest  of  $5420  for  17  months  at  4 
per  cent,  per  annum?  Ans.  $307.131. 

7.  What  is  the  interest  of  $7200  for  14  months  at  6 
per  cent,  per  annum?  Ans.  $504. 

8.  What  is  the  interest  of  $8050.871  for  3  years  and 
11  months  at  6  per  cent,  per  annum?  4ns.  $1891.95c.  5m. 

9.  What  is  the  interest  of  $948.621  for  8  months,  at 
8  per  cent,  per  annum?  Ans.  $50.59c.  3m. 

10.  What  is  the  interest  of  £421  16s.  9d.,  for  2  years 
and  8  months,  at  5  per  cent,  per  annum? 

Ans.  £56  4s. 


|32  SIMPLE    INTEREST. 

11.  What  is  the  interest  of  580  pounds  for  5  years,  % 
months  and  10  days,  at  7  per  cent,  per  annum? 

Ans.  £210.  17s.  lO^d. 

12.  What  is  the  interest  of  $36  for  1  month  at  8  per 
cent,  per  annum?  Ans.  24  cents. 

Bank  interest  is  generally  reckoned  by  days  only ;  and 
to  find  the  interest  for  any  number  of  days  at  6  per  cent, 
as  computed  at  banks,  multiply  the  dollars  by  the  num- 
ber of  days,  and  divide  by  6; — the  quotient  will  be  the 
interest  in  mills. 

Note.— The  interest  of  any  number  of  dollars  for  60 
days,  will  be  exactly  the  number  of  cents,  thus — $80 
for  60  days,  at  6  per  cent,  is  80  cents. 

CASE    III. 

The  amount,  time,  and  rate  per  cent,   given  to  find  the 
principal. 

RULE. 

Find  the  amount  of  100  dollars  at  the  rate  and  time 
given;  then  say,  as  the  amount  of  100  dollars,  is  to  the 
amount  given,  so  are  1 00  dollars  tr  the  principal  required. 

EXAMPLKS. 

1.  What  principal  at  interest  for  two  years,  at  6  per 
cent,  per  annum,  will  amount  to  $134.40? 
$100 
6 

6.00 

2 


12.00 
100.00 


$112  amount  of  100  for  two  years. 
dolls.         $  cts.        dolls,    dolls. 
Then,  as  112    :    134.40    :    :   100    :   120  Ans. 

2.  What   principal  at  interest  for  5  };ears,  at  6  per 
cent,  will  amount  to  $780?  Ans.  $600. 

3.  What  principal  at  interest  for  4  years  and  3  njonf  hs 
at  6  per  cent,  will  amount  to  $1192.25.  Ans.  $950? 


SIMPLE   INTEREST.  133 

CASE    IV. 

To  find  the  rate  per  cent,  when  the  amount,  time  and  prin- 
cipal are  given. 

RULE. 

Take  the  principal  from  the  amount,  the  remainder 
will  be  the  interest  for  the  given  time;  then,  as  the 
principal  is  to  one  hundred  dollars,  so  is  the  interest  of 
the  principal  for  the  given  time,  to  the  interest  of  100 
dollars  for  the  same  time.  Divide  the  interest  of  100 
dollars  thus  found,  by  the  time,  and  the  quotient  will  be 
the  rate  per  cent. 

EXAMPLES. 

1.  At  what  rate  per  cent,  will  $500  amount  to  $650 
in  three  years.  650  Amount. 

500  Principal. 

150  Interest  for  the  time, 
D.        D.          D       D. 
As  500   :   100   :    :   150  .  30  Interest  of  100. 
Then  divide  by  the  same  3)30(10  Ans.  per  cent 

30 

2.  At  what  rate  per  cent,  per  annum  will  $1850  dou- 
ble in  5  years?  Ans.  20  per  cent. 

CASE   V. 

To  find  the  time  when  the  principal,  amount,  and  rate  per 
cent,  are  given. 

RULE. 

Find  the  interest  of  the  principal  for  one  year;  find 
the  interest  of  the  principal  for  the  whole  time,  by  sub- 
tracting the  principal  from  the  amount;  then  divide  the 
whole  interest  by  the  interest  for  one  year — the  quotient 
will  show  the  time  required. 

EXAMPLES. 

1.  In  what  time  will  $300  amount  to  $1000  at  5  per 
cent,  per  annum? 

800  1000  Then,  4|0)20|0 

5  800  5 

$40|00  200  Whole  InH.    Ans.  5  years. 

12 


134  COMPOUND    INTEREST. 

2.  In  what  time  will  $80  amount  to  $182.40  at  8  per 
cent,  per  annum?  Ans.  16  years, 

COMPOUND    INTEREST. 

Compound  Interest  is  that  which  arises  from  the  in- 
terest being  added  to  the  principal,  and  becoming  a  part 
of  the  principal,  at  each  time  of  payment. 
RULE. 

Find  the  amount  of  the  principal,  for  the  time  of  the 
first  payment,  by  Simple  Interest;  this  amount,  contain- 
ing the  principal  and  interest  for  the  first  year,  will  be 
the  principal  for  the  second  year;  and  the  amount  of 
this  principal,  which  consists  of  the  principal  and  inte- 
rest for  the  second  year,  will  be  the  principal  for  the 
third  year,  and  so  on,  for  any  number  of  years.  From 
the  last  amount,  subtract  the  given  principal,  and  the 
remainder  will  be  the  compound  interest. 

EXAMPLES. 

1.  What  is  the  compound  interest  of  $8000  for  two 
years,  at  6  per  cent,  per  aunum? 

$8000 
6 

Interest  for  the  first  year  480100 
Principal  8000 

Amount  8480 
6 


Jfi't.  for  the  second  year  508.180 
Principal  8480.00 

8988.80 
Subtract  8000.00 

$988.  80c.  Answer. 

2.  What  is  the  compound  interest  of  $554  for  3  years, 
at  8  per  cent,  per  annum?  Ans.  $143.88. 

3.  What  is  the  compound  interest  of  $744  for  2  years, 
at  7  per  cent,  per  annum?  Ans.  $107.80c.  5m. 

4.  What  is  the  compound  interest  of  $50  for  8  years, 
$t  8  per  cent. per  annum?  Ans.  $42.54c.  6m, 


INSURANCE,   COMMISSION   AND   BROKERAGE.        133 

5.  What  is  the  compound  interest  of  £48  5s.  for  3 
years,  at  6  per  cent,  per  annum?  Ans.  £9  4s.  S^cL 

Q.   \.  What  is  interest? 

2.  What  are  the  four  things  considered  in  calculating 

interest? 

3.  What  is  the  principal ?-What  is  the  rate  per  cent.? 

What  is  the  time? — What  is  the  amount? 

4.  How  do  you  proceed  in  the  first  case? 

5.  How  do  you  proceed  in  pounds,  shillings,  &c.? 

6.  How  do  you  proceed  when  the  rt'te  per  cent,  con- 

sists  of  a  whole  number  and  a  fraction? 

7.  How  do  you  proceed  when  the  principal  is  in  dol- 

lars and  cents.? 

8.  How  do  you  calculate  interest  for  more  than  a 

year? — How,  when  the  time  is  in  months? 

9.  What  other  method  is  there  for  calculating  inte- 

rest, besides  the  method  of  multiplying  the 
sum  by  the  rate  p*r  cent.? 

10.  How  is  bank  interest  reckoned? — What  is  the  rule 

for  casting  it? 

11.  Do  you  understand  all  the  cases  and  rules  of  in- 

terest? 

12.  What  is  Compound  Interest? 

13.  Repeat  the  rule  for  calculating  Compound  Interest? 


INSURANCE,  COMMISSION  AND  BROKERAGE. 

Insurance,  Commission  and  Brokerage,  are  premiums 
allowed  to  insurers,  factors  and  brokers  at  a  certain  rate 
per  cent.;  and  is  obtained  after  the  manner  of  the  first 
case  in  Simple  Interest. 

EXAMPLES. 

1.  What  is  the  insurance  of  $4500,  at  2£  per  cent.*? 


9000 
2250 


$112l50c.  Answer, 


1 36  DISCOUNT* 

2.  What  is  the  commission  on  a  sale  of  goods  amotm 
ting  to  $1184  at  5  per  cent.?  Ans.  $59.2O, 

3.  What  is  the  brokerage  of  $987  at  3  per  cent.? 

Ans.  £99.61. 

4.  What  is  the  commission  on  a  sale  of  goods  amoun- 
ting to  $4820  at  4i  per  cent.?  Ans.  $21.6.90. 


DISCOUNT. 

Discount  is  an  allowance  made  for  the  payment  of 
any  sum  of  money  before  it  becomes  due,  and  is  the  dif- 
ference between  that  sum,  due  some  time  hence,  and  its 
present  worth.  RULE. 

As  the  amount  of  $100  at  the  given  rate  and  time  is 
to  $100,  so  is  the  given  sum  or  debt  to  the  present  worth. 
Subtract  the  present  worth  from  the  given  sum,  and  the 
remainder  will  be  the  discount. 

EXAMPLES. 

1 .  What  is  the  present  worth  of  $500  due  in  3  yearsr 
at  6  per  cent,  per  annum? 

$          $  $ 

$100  118   :   100   :   :  500 

6  100 

$    c.  m, 

6100  118)50000(423.72.8 
3  472 

18  280 

100  236 


118  amount  of  $100.  440 

354 


86.00|72c, 
82.6 

340 
236 

104  remainder. 


EQUATION.  1 37 

2.  What  is  the  present  worth  of  $350  payable  in  6 
months, discounting  at  6  per  cent,  per  annum? 

Ans.  $339  80c.  5m. 

3.  What  is  the  discount  on  $1000  due  in  one  year,  at 
6  per  cent,  per  annum?  Ans.  $56.60c.  3m. 

4.  What  is  the  present  worth  of  £65  due  in  15  months 
at  6  per  cent,  annum?  Ans.  £60  9s.  3£d. 

5.  What  sum  will  discharge  a  debt  of  $1695  due  af- 
ter 5  months  and  20  days  at  6  per  cent,  per  annum? 

Ans.  $1541.32c.  6m. 

6.  What  is  the  present  worth  of  $426.55,  at  6  per 
cent,  per  annum,  due  in  8  months?     Ans.  $410.14c.  5m. 

Note. — When  discount  is  made  without  regard  to  time, 
it  is  found  as  the  interest  of  the  sum  would  be  for  one 
year. 


EQUATION. 

Equation  is  the  method  for  finding  a  time  to  pay  at 
once,  several  debts  due  at  different  times. 
RULE. 

Multiply  each  payment  by  the  time  at  which  it  is  due, 
and  divide  the  sum  of  the  products  by  the  sum  of  all  the 
payments — the  quotient  will  be  the  time  required. 

EXAMPLES. 

1.  A  owes  B  $480  to  be  paid  in  the  following  manner, 
viz:  $100  in  6  months,  $120  in  7  months,  and  $260  in  10 
months;  what  is  the  equated  time  for  payment  of  the 
whole  debfc? 

100X  6=    600 

120X  7=    840 

260X10=  2600 


480  )4040(8T5g  months, 

3840 


138  LOSS    AND    GAIN. 

2.  A  owes  B  $1100,  of  which  200  is  to  be  paid  in  3 
months,  400  in  5  months,  and  500  in  8  months — what  is 
the  equated  time  for  payment  of  all?       Ans.  6  months. 

3.  C  is  indebted  to  a  merchant  to  the  amount  of  $2500; 
of  which  $1000  is  payable  at  the  end  of  4  moMths,$800 
in  8  months,  and  700  in   12  months — when  ought  pay- 
ment to  be  made,  if  all  are  paid  together? 

Ans.  7  months  15J  days. 


LOSS  AND  GAIN. 

Loss  and  Gain  is  a  rule  by  which  persons  in  trade  are 
able  to  discover  their  profit  or  loss;  and  to  increase  or 
lessen  the  prices  of  their  goods  so  as  to  gain  or  lose  on 
them  to  any  given  amount. 

Questions  in  Loss  and  Gain  are  solved  by  the  Rule  of 
Three,  or  by  Practice. 

EXAMPLES. 

1.  A  merchant  bought  100  yards  of  silk  at  75  cents 
per  yard,  what  will  be  his  gain  in  the  sale,  if  he  sell  it 
at  90  cents  per  yard. 

75  cents. 

yard,  yards.      cts.    Dolls. 
15  gain  per  yard.      As  1    :  100   :   :   15   :  15  Ans. 

2.  If  a  grocer  buy  250  Ibs.  of  tea,  at  $225,  and  sell 
the  whole  at  $1.25  per  Ib.  what  will  be  his  gain  by  the 
transaction? 

$1.25  $312.50 

250  225.00 


G250  $87.50 

250 


$312.50 

3.  If  a  yard  of  calico  cost  28  cents,  and  is  sold  for  31 
cents,  what  is  the  gain  on  293  yards?  Ans.  $8.79. 

4.  Bought  420  bushels  of  corn  at  25  cents  per  bushel, 
and  sold  the  same  at  38  cents  per  bushel;  what  was  the 
amount  gained?  Ans.  $54.60. 


INVOLUTION.  139 

5.  A  merchant  bought   12  cwt.  of  coffee  at  26  cents 
per  Ib,  and  afterwards  obliged  to  sell  it  20  cents  per  Ib. 
what  was  his  loss?  Ans.  $00.64. 

6.  It  a  merchant  gain  $80  on  $560,  what  is  that  per 
cent.?  Ans   14f  per  cent. 

7.  If  a  yard  of  vel  vet  be  bought  for  1 6s.  and  sold  again 
for  12s.  what  is  the  loss  per  cent.?        Ans.  25  per  cent, 


INVOLUTION, 

OR   THE    RAISING   OF    POWERS* 

The  product  of  any  number  multiplied  by  itself  any 
given  number  of  times,  is  called  its  power,  as  in  the  fol- 
lowing example. 

Thus,     2X2  =  4  the  square,  or  second  power  of  2. 

2X2X2  =  8  the  cube,  or  third  power  of  2. 
2X2X2X2=  16  the  biquadrate,  or  4th  power  of 
2.*     Hence,  3  raised  to  the  4th  power  makes  81.     The 
number  which  denotes  a  power  is  called  the  index,  or 
exponent  of  that  power. 

When  a  power  of  a  vulgar  fraction  is  required,  it  is 
only  necessary  to  raise,  first  the  numerator,  and  then  the 
denominator  to  the  given  power,  and  place  the  product 
of  the  one  over  the  product  of  the  other;  thus,  the  3d 

2X2X2  == 
power  of  *  3X3X3  =  *' 

EXAMPLES. 

1.  What  is  the  square  of  4567?  Ans.  20857489. 

2.  What  is  the  cube  of  567?  Ans.  182284263, 

3.  What  is  the  biquadrate  of  67?        Ans.  20151121. 

4.  What  is  the  ninth  power  of  2?  Ans.  512. 

5.  What  is  the  cube  of  J?  Ans.  |ff. 

6.  What  is  the  cube  or  third  power  of  .13? 

Ans.  .002197. 

7.  What  is  the  sixth  power  of  5.03? 

Ans.  16196.005304479729, 

*  Any  given  number  is  considered  the  first  power  of  itself,  and  when  multiplied 
by  itself  the  product  is  the  second  power,  &c. 


HO  E.V0LUT10N. 

TABLE    OF   THE    FIRST    IVINE   POWERS. 


? 

o 

C/3 

-0 
fi 

1 
1 

O 
c 
cr 

O> 

£» 
S" 

T3 
0 

<T> 

in 

cr 
"3 

0 
<T> 

^t 

a* 

i 

o 

-* 

cr 

T3 
0 

<B 

CO 

tr 
*d 

0 

n 

CO 

& 

1 

0 

p 

1 

1 

1 

16 

1 

1 

1 

1 

1 

2 

4 

8 

«->!i!. 

64 

128 

256 

512 

n 

O 

9 

27 

SI 

241' 

729 

2187 

6561 

19683 

4 

16 

64 

256 

1024 

4096 

16384 

65536 

262144 

5 

25 

E 
f, 

125 

625 

3125 

15625 

78125 

390625 

1953125 

6 

7 
8 
9 

216 

1296 

7776 
1680? 

46656 

279936 

1979616 

10077696 

342? 

5401 

117649 

823543 

5764801 

40353607 

64 

512 

4096 

-J2768 

>621  44  2097152  16777216 

134217728 

81 

72C 

6561 

o904fJ 

531441  14782969S43046721 

38742048S 

EVOLUTION, 

OR   THE    EXTRACTION   OF   ROOTS. 

The  root  of  a  number,  or  power,  is  any  number,  which 
Ueing  multiplied  by  itself  a  certain  number  of  times, 
will  produce  that  power;  and  is  called  the  square,  cube, 
biquadrate  root,  &c.  according  to  the  power  to  which  it 
belongs.  Thus,  3  is  the  square  root  of  9,  because  when 
multiplied  by  itself,  it  produces  9;  and  4  is  the  cube 
root  of  64,  because  4X4X4  =  643  and  so  of  any  other 
number. 


THE    SQUARE   ROOT. 

Extracting  the  square  root  of  a  number,  is  the  taking 
a  smaller  number  from  a  larger,  and  *uch  as  will,  being 
multiplied  by  itself,  produce  the  larger  number, 


EVOLUTION.  141 

RULE. 

1.  Separate  the  sum  into  periods  of  two  figures  each, 
beginning  at  the  right  hand  figure. 

2.  Seek  the  greatest  square  number  in  the  left  hand 
period;  place  the  square, thus  found, under  that  period, 
and  the  root  of  it  in  the  quotient.     Subtract  the  square 
number  from  the  first  period;  to  the  remainder  bring 
down  the  next  period,  and  call  that  the  resolvend. 

3.  Double  the  quotient,  and  place  it  on  the  left  hand 
of  the  resolvend  for  a  divisor.     Seek  how  often  the  di- 
visor is  contained  in  the  resolvend,  omitting  the  units 
figure,  and  set  the  answer  in  the  quotient,  and  also  on 
the  right  hand  side  of  the  divisor.     Then  multiply  the 
divisor,  including  the  last  added  figure,  by  that  figure, 
that  is,  by  the  figure  last  placed  in  the  quotient;  place 
the  product  under  the  resolvend,  subtract  it,  and  to  the 
remainder  bring  down  the  next  period,  if  there  be  any 
more,  and  proceed  as  already  directed.     If  there  be  a 
remainder  after  the  periods  are  all  brought  down,  annex 
cyphers,  two  at  a  time,  for  decimals,  and  proceed  till 
the  root  is  obtained  with  sufficient  exactness. 

Note. — When  a  sum  in  the  Square  Root  consists  of 
whole  numbers  and  decimals,  point  off  the  whole  num- 
bers as  above  directed,  then  point  the  decimal  part, 
commencing  at  the  decimal  point  and  forming  periods  of 
two  figures  each  towards  the  right,  observing  when 
there  is  only  one  figure  left  for  the  last  period,  to  add  a 
cypher  to  the  right  of  it,  to  make  an  even  period. — 
When  the  sum  consists  entirely  of  decimals,  separate 
the  periods  after  the  same  manner.  If  it  be  required 
to  extract  the  square  root  of  a  vulgar  fraction,  reduce 
it  to  its  lowest  terms;  then  extract  the  root  of  the  nu- 
merator for  the  numerator  vf  the  answer,  and  the  root 
of  the  denominator  for  the  denominator  of  the  answer. 
If  the  fraction  be  a  surd,  that  is,  a  number  whose  root 
can  never  be  exactly  found,  reduce  it  to  a  decimal,  and 
then  extract  the  root  from  it;  and  if  the  sum  be  a  mixed 
number,  the  root  may  be  obtained  in  the  same  way. 
PROOF. 

Square  the  root,  adding  in  the  remaider,  (if  any,)  and 
the  result  will  equal  the  given  number. 


142 


EXAMPLES. 

1.  Whaft  is  the  square  root  of  20857489"? 

....  Root. 
20857489(4567  Answer, 
16 

divisor  85)485  resolvend. 
425 


divisor  906)6074  resolvend. 
5436 

divisor  9127)63889  resolvend. 
63889 

2.  What  is  the  square  root  of  294849?          Ans.  54$. 

3.  What  is  the  square  root  of  41242084?    Aris.  6422, 

4.  What  is  the  square  r^ot  of  17.3056?        Ans.  4.16, 

5.  What  is  the  square  root  of  .000729?        Ans.  .027, 

6.  What  is  the  square  root  of  5?  Ans.  2.23606, 

7.  What  is  the  square  root  of  //T?  Ans.  ^. 

8.  What  is  the  square  root  of  17f  ?       Ans.  4.168333. 

9.  A  general  has  an  arrny  of  7056  men;  how  many 
must  he  place  on  a  side  to  form  them  into  a  compact 
square?  Ans.  84. 

10.  If  the  area  of  a  circle  he  184.125,  what  is  the  side 
of  a  square  that  shall  contain  the  same  area? 

Thus,  ^184.125=  13.569-|-Answer. 

11.  If  a  square  piece  of  land  contain  61  acres  and  41 
square  poles,  what  is  the  length  of  one  of  its  sides? 

A.  P. 

Thus,  61   41  =9801  square  poles. 
Then,  ^9801  =  99  rods,  or  poles  in  length,  Answer. 

12.  There  is  a  circle  whose  diameter  is  4  inches;  what 
is  the  diameter  of  a  circle  3  times  as  large? 

Thus,  4X4=  16;  and  16X3=  48  and  ^48=  6.928 
-{-inches.     Ans. 

13.  There  is  a  circle  whose  diameter  is  8  inches;  what 
is  the  diameter  of  a  circle  which  is  only  one  fourth  as 
farge.    8X8=64;  and  64-7-4=16;  and  ^16=4  inches, 

Ans,  4  inches, 


THE   CUBE   ROOT.  143 

The  square  of  the  longest  side  of  a  right  angled  trian- 
rJe,  is  equal  to  the  sum  of  the  squares  of  the  other  two  sides; 
therefore,  the  difference  of  the  squares  of  the  longest  side, 
and  either  of  the  other  sides,  is  the  square  of  the  remaining 
side. 

14.  The  wall  of  a  certain  city  is  20  feet  in  height,  it 
is  surrounded  by  a  ditch  20  feet  in  breadth;  what  must 
be  the  length  of  a  ladder,  to  reach  from  the  outside  of 
the  ditch  to  the  top  of  the  wall?  Ans.  28i  feet. 


THE  CUBE  ROOT. 

The  cube  root  of  a  given  number,  is  such  a  number 
as  being  multiplied  by  itself,  and  then  into  that  product, 
preduces  the  given  number. 

RULE. 

1.  Point  off  the  sum  into  periods  of  three  figures  each, 
beginning  with  units. 

2.  Find  the  greatest  cube  in  the  left  hand  period, 
place  the  root  of  it  in  the  quotient,  subtract  the  cube 
from  the  left  hand  period,  and  to  the  remainder  bring 
down  the  next  period  for  a  resolvend. 

3.  Square  the  quotient,  and  multiply  the  square  by  3 
for  a  defective  divisor. 

4.  Seek  how  often  the  defective  divisor  is  contained 
in  the  resolvend,  omitting  the  units  and  tens,  or  two 
right  hand  figures.     Place  the  result  in  the  quotient,  and 
its  square  to  the  right  of  the  divisor,  supplying  the  place 
of  tens  with  a  cypher,  whenever  the  square  is  less  than 
ten. 

5.  Multiply  the  last  figure  of  the  quotient  or  root  by 
all  the  figures  in  it  previously  ascertained ;  multiply  that 
product  by  30,  and  add  their  product  to  the  divisor,  to 
complete  it. 

6.  Multiply  and  subtract  as  in  Simple   Division,  and 
to  the  remainder  bring  down  the  next  period,  for  a  new 
resolvend.     Find  a  divisor  as  before,  and  thus  proceed 
imtil  all  the  periods  are  brought  down. 


}44  THE    CUBE    ROOT. 

Note.  —  The  cube  root  of  a  vulgar  fraction  is  found  by 
reducing  it  to  its  lowest  terms,  and  extracting,  as  in  the 
square  root;  and  if  the  fraction  be  a  surd,  reduce  it  to 
a  decimal,  and  then  extract  the  root. 

In  extracting  the  cube  root,  if  the  sum  be  in  part  de- 
cimals, or  if  the  whole  be  decimals,  point  the  figures  as 
in  the  square  root,  observing  to  have  three  figures  in  a 
period  instead  of  two;  and  in  all  cases  in  the  cube  root, 
when  there  is  a  remainder,  if  it  be  required  to  obtain 
decimal  figures  to  the  root,  proceed  as  directed  in  the 
square  root,  only  add  three  cyphers,  in  place  of  two,  to 
the  remaider. 

PROOF. 

Involve  the  root  to  the  third  power,  adding  the  re- 
mainder, (if  any,)  to  the  result. 

EXAMPLES. 

1.  What  is  the  cube  root  of  182284263? 

.      .      .  Root. 
182284263(567  Answer. 
125 
f  Defective  divisor  and  square  of  6.  - 


Complete  divisor  —  8436)50616 


Defective  56X56X3  =940849),™™^ 
divisor.     7X56X30=   11760J6^8263  new  resolv. 

Complete  divisor     952609)6668263 

2.  What  is  the  cube  root  of  48228  544?       Ans.  36.4. 

3.  What  is  the  cube  root  (or  3d  root)  of  2? 

Ans.  1.259921. 

4.  What  is  the  cube  root  of  132651  ?  Ans.  51. 

5.  What  is  the  cube  root  of  4173281?  ADS.  161. 

6.  What  is  the  cube  root  of  .008649?     Ans.  .2052-f. 

7.  What  is  the  cube  of  iff?  Ans.  f. 

8.  If  the  contents  of  H  globe  amount  to  5832  cubick 
inches,  what  are  the  dimensions  of  the  side  of  a  cubick 
block  containing  the  same  quantity  ?     Ans.  1 8  in.  square 


BIftUADRATE   ROOT.  14'i> 

THE  BiaUADRATE  ROOT. 

To  extract  the  biquadrate  root,  is  to  find  out  a  number 
which  being  involved  4  times  into  itself,  will  produce 
the  given  number,  that  is  the  fourth  power. 
RULE. 

Extract  the  square  root  of  the  sum,  then  extract  the 
square  root  of  that  root,  and  the  last  root  will  be  the 
answer. 

EXAMPLES. 

1  What  is  the  biquadrate  root  (or  4th  root)  of  531441? 

Root. 

531441  )  729  (  27  Answer, 
49  4 

142)414       47)329 
284  329 


1449)13041 
13041 


2.  What  is  the  biquadrate  root  of  4096?          Ans.  *. 

3.  What  is  the  biquadrate  root  of  146841?     Ans.  11. 
Mote. — The  roots  of  several  other  powers  may  be  ob- 
tained by  means  of  the  foregoing  rules,  thus — 

To  obtain  the  root  of  the  6th  power,  extract  the  square 
root  of  the  cube  root. 

For  the  8th,  take  the  square  root  of  the  biquadrate 
root.  For  the  9th,  take  the  cube  root  of  the  cube  root. 
For  the  12th  root,  take  the  cube  root  of  the  biquadrate 
root. 

Questions  concerning  the  powers  and  roots. 

1.  WThat  is  called  a  power? 

2.  AVhat  power  is  the  square?    Ans.  The  2d.  power 

3.  What  is  the  cube  of  a  number  called? 

4.  AVhat  is  the  biquadrate? 

5.  How  do  you  raise  the  pow^r  of  a  vulgar  fraction  I 

6.  What  is  the  root  of  a  power? 

7.  What  is  meant  by  extracting  the  square  root'? 

8.  Repeat  the  rule  for  doing  it? 


146  ALLIGATIONS7. 

9.  How  do  you  proceed  when  the  sum  consists  in  patt; 
or  altogether,  of  decimals? 

10.  How  do  you  extract  the  square  root  of  a  vulgar 

fraction  ? 

11.  How  do  you  proceed  when  the  fraction  is  a  surd? 

12.  What  do  you  understand  by  the  cube  root? 

13.  Repeat  the  rule  for  extracting  it? 

14.  How  do  you  extract  the  cube  root  of  a  vulgar  frac- 

tion? 

15.  What  do  you  understand  by  extracting  the  biqua- 

drate  root? 

16.  Repeat  the  rule  for  extracting  it? 

17.  How  are  sums  in  the  square  roet  proved? 

18.  How  are  sums  in  the  cube  root  proved.? 


ALLIGATION. 

Alligation  is  a  rule  for  mixing  simples  of  different 
qualities,  in  such  a  manner  that  the  composition  may  be 
of  a  mean  or  middle  quality. 

CASE   I. 

To  find  the  mean  price  of  any  part  of  the  mixture,  when 
the  quantities  and  prices  of  several  thing*  are  given. 

RULE. 

As  the  sum  of  the  quantities  is  to  any  part  of  the  com- 
position, so  is  the  price  of  the  quantities  to  the  price  of 
any  particular  part. 

EXAMPLES. 

1.  A  trader  mixes  60  gallons  of  wine  at  100  cents  per 
gallon;  40  gallons  at  80  cents,  and  30  gallons  of  water. 
What  should  be  the  price  per  gallon? 

gals.     cts.        % 
Wine     60  at  100=60.00 
Wine     40  at    80  =  32.00 
Water  30 

gals.    gal.          $ 
J30        1     :   :  92.0* 


ALLIGATION.  1 4? 

$.  A  trader  mixes  a  quantity  of  tea  as  follows,  viz:— 
*lbs.  of  tea  at  42  cents  per  lb.;  6  Ibs.  at  33  cents;  121bs, 
75  cents,  and  15  IDS.  at  30  cents.  What  can  he  sell  it 
for  per  lb.?  Ans.*66ff  cents, 

3.  A  farmer  mixes  20  bushels  of  wheat  at  5s.  per 
bushel,  with  36  bushels  of  rye  at  3s.,  and  40  bushels  of 
of  barley  at  2s,  per  bushel;  how  much  is  a  bushel  of 
*,he  mixture  worth?  Ans.  3s, 

CASE    II. 

When  the  prices  of  several  simples  are  given  to  find  what 
quantity  of  each,  at  their  respective  prices,  must  he  taken 
to  make  a  compound  at  a  proposed  price. 

RULE. 

Set  the  prices  of  the  simples  in  a  column  under  each 
other.  Connect  with  a  continued  line,  the  rate  of  each 
simple  which  is  less  than  that  of  the  compound,  with  one 
or  any  number  of  those  that  are  greater  than  the  com- 
pound, and  each  greater  rate,  with  one  or  more  of  the 
less.  Place  the  difference  between  the  mixture  rate, 
and  that  of  each  of  the  simples,  opposite  to  the  rates 
with  which  they  are  linked.  Then,  if  only  one  differ- 
ence stand  against  any  rate,  it  will  be  the  quantity  be- 
longing to  that  rate;  but  if  there  be  several,  their  sum 
will  be  the  quantity.  Different  modes  of  linking  will 
produce  different  answers. 

EXAMPLES. 

1.  A  merchant  would  mix  wines  at  17s.  18s.  and  22s, 
per  gallon,  so  that  the  mixture  may  be  worth  20s.  per 
gallon:  what  quantity  of  each  must  be  taken? 


Ans.  2  gallons  at  17s.,  2  at  18s.,  and  5  at  22s. 

2.  How  much  barley  at  40  cents, corn  at  60, and  wheat 

at  80  cents  per  bushel,  must  be  mixed  together,  that  the 

compound  may  be  worth  62|  cents  per  bushel? 

Ans.  17J-  bush,  of  barley,  17|-  of  corn,  and  25  of  wheat; 


14$  ALLIGATION- 

CASE    III. 

When  the  prices  of  all  the  simples,  the  quantity  of  one  of 
them,  and  the  mean  price  of  the  mixture,  are  given,  to 
.find  the  quantities  of  the  other  simples. 

RULE. 

Find  an  answer  as  before, by  connecting;  then,  as  the 
difference  of  the  same  denomination  with  the  given 
quantity,  is  to  the  differences  respectively,  so  is  the  given 
quantity,  to  the  different  quantities  required. 

EXAMPLES. 

1.  How  much  gold  of  15,  17,  18,  and  22  carats  fine 
must  be  mixed  together  to  form  a  composition  of  40  oz> 
of  20  carats  fine? 

f!5  ^  .         .         2 

Mean  or  Mixture  j  17^  2 

rate.     20      |  IS\  \  2 

L22;J  5+3+2=10 


16 
then  as  16    :     2   :    :  40   :     5) 


and  as   16   :  10   :    :  40   :  25J  AnSW6r' 
Answer  5  oz.  of  15, 17,  and  18  carats  fine,  and  25  QXI 
of  22  carats  fine. 

2.  A  grocer  has  currents  at  4d.,  6d.,  9d.,  and  lid., 
perlb.  and  he  would  make  a  mixture  of  240  Ibs.  that 
might  be  sold  at  8d.  perlb.j  how  much  of  each  kind- 
must  he  take? 

Ans,  72 Ibs.  at  4d.,  24  at  6d.,  48  at  9d.  and  96  at  lid, 

CASE   IV. 

When  the  prices  of  the  simples,  the  quantity  to  be  mixed, 
and  the  mean  price  are  given,  to  find  the  quantity  of  each 
simple. 

RULE* 

Connect  the  several  prices,  and  place  their  differen- 
ces as  before;  then,  as  the  sum  of  the  differences  thus 
given,  is  to  the  difference  of  each  rate,  so  is  the  quantl 
ty  to  be  compounded,  to  the  quantity  required 


POSITION*  149 

EXAMPLES. 

1.  How  much  sugar  at  9  cents,  11  cents  and  14  cents 
per  Ib.  will  be  necessary  to  form  a  mixture  of  20  Ibs, 
worth  12  cents  perlb.? 

(91  2 

12  Iliv  |  2 

(14/J  3+1  =  4 

8 
Then,  as  8      2   :  :  20   :    5  Ibs.     9  cents.) 


2   :   :  20   :    5  Ibs.  11  cents.)  Answer. 


4   *   :  20   :  10  Ibs.  14  cents.) 

2.  A  grocer  has  sugar  at  24  cents  perlb.  and  at  13 
cents  perlb.;  and  he  wishes  so  to  mix  2cwt.  of  it,  that 
he  may  sell  it  at  16  cents  perlb.;  how  much  of  each 
kind  must  he  take?     Ans.  162j£  Ibs.  of  that  at  13  cents, 
and  61TVlbs.  of  that  at  24  cents. 

3.  How  many  gallons  of  water  must  be  mixed  with 
wine  worth  60  cents  per  gallon,  so  as  to  fill  a  vessel  of 
80  gallons,  that  may  be  sold  at  41  £  cents  per  gallon? 

Ans.  18f  gallons  of  water,  and  61  A  of  wine> 


POSITION. 

Position  is  a  rule  for  solving  questions,  by  one  or  more 
supposed  numbers.  It  is  divided  into  two  parts,  namely 
single  and  double, 


SINGLE   POSITION. 

Single  position  teaches  to  solve  questions  which  re- 
quire but  one  supposition. 

RULE. 

Suppose  a  number,  and  proceed  with  it  as  if  it  were 
the  real  one,  setting  down  the  result — Then,  as  the  re- 
Suit  of  that  operation,  is  to  the  number  given,  so  is  the 
supposed  number,  to  the  number  sought, 
IS* 


1 50 

EXAMPLES. 

1.  What  number  is  that,  which  being  multiplied  by 
and  the  product  divided  by  6,  will  give   14  for  the 
flentf 

Suppose  18 

7 

6)126 

Theft,  as  21    :   14  :   :  1& 
18 

112 
14 

£  1)252(12  Answer 
21 

42 
42 

&  What  number  is  that,  of  which  one  half  eiceeefe 
^ne  third  by  15? 

Suppose  60— Then  £  |  60  |  i|60 

30        20 
Subtract  20 

10 
rtfhen,  as  10  :  15   :   :  60   :  90  Answer. 

3.  What  number  is  that,  which  being  increased  by  |? 
•J.  and  J  of  itself,  the  sum  will  be  125?  Ans.  60, 

4.  A  schoolmaster  being  asked  how  many  scholars  he 
foad,  answered,  that  if  |  of  his  number  were  multiplied 
by  7,  and  |  of  the  same  number  added  to  the  product, 
the  sum  would  be  292.     What  was  his  number?  Ans.  60. 

5.  A  schoolmaster  being  asked  what  number  of  schol- 
ars he  had,  said,  if  I  had  as  many,  half  as  many,  and 

fourth  as  many,  I  should  have  99.     What  was  hrs 

Ans,  -36 


POSITION.  1^> 

b.  A  person,  after  spending  i  and  £  of  his  money. v 
§30  left;  what  had  he  at  first?  Ans.  $180. 

7.  Seven  eighths  of  a  certain  numher  exceed  four 
fifths  by  6.     What  is  that  number?  Ans.  80. 

8.  A  certain  sum  of  money  is  to  be  divided  among  4 
persons,  in  such  a  manner  that  the  first  shall  have  i  of 
it,  the  second  |,  the  third  £,  and  the  fourth  the  remain- 
der, which  is  $28;  what  is  the  sum?  Ans.  $112. 

9.  What  sum,  at  6  per  cent,  per  annum,  will  amount 
t®  £860.  in  12  years?  As. 


DOUBLE   POSITION. 

Double  Position  teaches  to  resolve  questions  by  meaifs 
of  two  supposed  numbers. 

RULE. 

Suppose  two  convenient  numbers,  and  proceed  with 
each  according  to  the  condition  of  the  question,  and  set 
down  the  efrours  of  the  results.  Multiply  the  errours 
into  their  supposed  numbers,  crosswise ;  that  is,  multiply 
the  first  supposed  number  by  the  last  errour,  and  the 
last  supposed  number  by  the  first  errour. 

If  the  errours  be  alike,  that  is,  both  too  much,  or  both 
too  little,  divide  the  difference  of  their  products  by  the 
difference  of  the  errours — the  quotient  will  be  the  an- 
swer. But  if  the  errours  be  unlike,  that  is,  one  too- 
large  and  the  other  too  small,  divide  the  sum  of  the 
products  by  the  sum  of  the  errours. 

EXAMPLES. 

1.  What  number  is  thatj  whose  1  part  exceeds  the  | 
part  by  16? 

Suppose  24;  and  as  1  of  24  is  8,  and  1  of  it  is  6,  it  is 
evident  that  the  third  part  exceeds  the  fourth  part  by  2f 
instead  of  16;  and  therefore  the  errour  is  14  too  small, 
Again,  suppose  48;  and  i  of  48  being  16,  and  i  being  12, 
it  is  manifest  that  the  third  part  exceeds  the  fourth  by 
4,  instead  of  16;  hence  the  errour  is  12  too  small,—*-- 
Then,  the  errours  being  alike,  proceed  thus— » 


POSITION, 

er. 
1.  supposition  24  \          /  14  too  small, 


er. 
2.  supposition  48  /       \  12  too  small, 


14  672  product.  288  product, 

12  288 

2  dif.of  er.2)384  difference  of  the  products. 

192  Answer. 

2.  A  son  asking  his  father  how  old  he  was,  received 
this  answer:    Your  age  is  now  '-  of  mine;  but  5  years 
ago,  your  age  was  |  of  mine.     What  are  their  ages? 

Ans.  20  and  80. 

3.  Two  persons,  A  and  B,  have  each  the  same  income, 
A  saves  \  of  his;  but  B,  by  spending  50  dollars  per  an- 
num more  than  A,  finds  himself  at  the  end  of  4  years 
one  hundred  dollars  in  debt.     What  was  their  income, 
and  what  did  each  spend? 

Ans.  Their  income  was  $125  per  annum  for  each,-  A 
spends  $100  and  B  spends  $150  per  annum. 

4.  What  number,  added  to  the  sixty-second  part  of 
7626,  will  make  the  sum  of  200?  Ans.  77. 

5.  A  man  being  asked  how  many  sheep  he  had  in  his 
drove,  said,  if  1  had  as  many  more,  half  as  many  more, 
one  fourth  as  many  more,  and  12|,  I  should  have  40. — 
How  many  had  he?  Ans.  10. 

6.  An  officer  had  a  divison,  £  of  which  consisted  of 
English  soldiers,  \  of  Irish,  £  of  Canadians,  and  50  of 
Ihdians.     How  many  were  there  in  the  whole?  Ans.  600. 

7.  A  servant  being  hired  for  30  days,  agreea  to  re- 
ceive 2s.  6d.  for  every  day  he  laboured,  and  to  forfeit 
Is.  for  every  day  he  played.     At  the  end  of  the  term 
his  pay  amounted  to  £2.  14s.     How  many  of  the  days 
did  he  labour?  Ans.  24. 

8.  What  number  is  that,  which  being  multiplied  by  6, 
the  product  increased  by  adding  18  to  it,  and  the  sum 
divided  by  9,  the  quotient  will  be  20?  Ans,  27, 


ARITHMETICAL   PROGRESSION".  153 

ARITHMETICAL  PROGRESSION. 

Arithmetical  Progression  is  a  series  of  numbers  in< 
creasing  or  decreasing  by  a  common  difference;  as,  1^ 
2,  3,  4,  5;  1,  3,  5,  7,  9;  5,  4,  3,  2,  1;  9,  7,  5,  3,  1,  &c. 
The  numbers  in  a  series  are  called  terms — the  first  and 
last  terms  are  called  extremes,  and  the  common  differ- 
ence is  the  number  by  which  the  terms  in  a  series  differ 
from  each  other;  as  in  2,  5,  8,  11,  &c. — the  common  dif- 
ference is  3. 

In  any  series  in  Arithmetical  Progression,  the  sum  of 
the  two  extremes  is  equal  to  the  sum  of  any  two  terms, 
equally  distant  from  them,  or  equal  to  double  the  mid- 
dle term  when  there  is  an  uneven  number  of  terms  in 
the  series.  Thus,  in  the  series  2,  4,  6,  8,  10,  12, — the 
extremes  are  2  and  12,  equal  to  14,  and  if  you  add  10 
and  4,  or  8  and  6,  the  result  will  be  the  same;  and  in  the 
series  2,  4,  6,  8,  10,  the  extremes  are  10  and  2,  and  as 
the  number  of  terms  is  uneven  6  is  the  middle  one, 
which,  when  doubled  makes  12,  and  the  extremes  whea 
added  together  make  the  same  amount, 

CASE   I. 

The  first  term,  common  difference,  and"  number  of  terms, 
being  given,  to  find  the  last  term  and  sum  of  all  the 
terms. 

RULE. 

Multiply  the  common  difference  by  one  less  than  the 
number  of  terms,  and  to  the  product  add  the  first  term, 
the  sum  will  be  the  last.  Add  the  first  and  last  terms 
together,  multiply  their  sum  by  the  number  of  terms, 
and  half  the  product  will  be  the  sum  of  all  the  terms. 

EXAMPLES. 

1.  The  first  term  in  a  certain  series  is  3,  the  common 
difference  2,  and  the  number  of  terms  9;  to  find  the  last 
ferm,  and  the  sum  of  all  the  terms, 


154  ARITHMETICAL    PROGRESSION 

One  less  than  the  number  of  terms  is  8. 
2  common  difference. 

8  number  of  terms  less  oiie 

16  product. 
3+first  term 

19  last  term. 
3+first  term. 

22 

9  X number  of  terms. 

2)198 

Answer    99  sum  of  all  the  terms. 
£.  A  person  sold  80  yards  of  cloth  at  3  cents  for  th€ 
Urst  yard.  6  for  the  second,  and  thus  increasing  3  cents 
every  yard :  what  was  the  whole  amount?    Ans.  $97.20. 

3.  How  many  times  does  a  clock  usually  strike  in  12 
hours?  Ans.  78. 

4.  A  man  on  a  journey  travelled  20  miles  the  first 
day.  24  the  second,  and  continued  to  increase  the  num- 
ber of  miles  by  every  clay  for  10  day.     How  far  did  he 
travel?  Ans.  380  miles. 

5.  A  farmer  bought  20  cows,  and  gave  2  dollars  for 
the  first,  4  for  the  second,  and  so  on,  giving  in  the  same 
proportion  from  the  first  to  the  last.     What  did  he  give 
for  the  whole?  Ans.  $420. 

CASE   II. 

When  the  two  extremes  and  the  number  of  terms  are  given 
to  find  the  common  difference. 

RULE. 

Subtract  the  less  extreme  from  the  greater,  and  di- 
vide the  remainder  by  one  less  than  the  number  of 
terms — the  quotient  will  be  the  common  difference. 

EXAMPLES. 

1.  The  extremes  being  3  and  19,  and  the  number  of 
ferms  9,  what  is  the  common  difference? 


GEOMETRICAL   PROGRESSION.  ^5 

9  19  II. 

13  13  number         "\ 

1  of  terms  80  '  p  . 
8       8)16  -  2Qf  Extremes. 

12  -J 

Ans.  2  12)60  difference  of 

—  extremes. 
Common  difference  5  Answer. 

3.  If  the  extremes  be  10  and  70,  and  the  number  of 
terms  21,  what  is  the  common  difference,  and  the  sum 
of  the  series?  Ans.  com.  diff.  3,  and  the  sum,  840. 

4.  A  certain  debt  ctm  be  paid   in  one  year,  or  52 
weeks,  by  weekly  payments  in  Arithmetical  Progres- 
sion, the  first  payment  being  1  dollar,  and  the  last  103 
dollars.     What  is  the  common  difference  of  the  terms? 

Ans.  $2. 

5.  A  debt  is  to  be  discharged  at  16  several  payments 
in  Arithmetical  Progression;  the  first  payment  to  be  20 
dollars,  and  the  last  1 10  dollars.     What  is  the  common 
difference.  Ans.  $6, 


GEOMETRICAL  PROGRESSION. 

Geometrical  Progression  is  the  increase  of  any  series 
of  numbers  by  a  common  multiplier,  or  the  decrease  of 
any  series  by  a  common  divisor;  as  3,  6,  12,  24,  48; 
and  48,  24,  12,  6, 3.  The  multiplier  or  divisor  by  which 
any  series  is  increased  or  decreased,  is  called  the  ratio. 

CASE    I. 

To  find  the  last  term  and  sum  of  the  series, 
RULE. 

Raise  the  ratio  to  a  power  whose  index  is  one  less 
than  the  number  of  terms  given  in  the  sum.  Multiply 
the  product  by  the  first  term,  and  the  product  of  that 
multiplication  will  be  the  last  term:  then  multiply  the 
last  term  by  the  ratio,  subtract  the  first  term  from  the 
product,  and  divide  the  remainder  by  a  number  that  is 
one  less  than  the  ratio — the  quotient  will  be  the  sum  of 
^he  series. 


156  GEOMETRICAL   PROGRESSION. 

EXAMPLES. 

1.  Bought  12  yards  of  calico,  at  2  cents  for  the  first 
yard,  4  cents  for  the  second,  8  for  the  third,  &c.  :  what 
was  the  whole  cost? 

NOTE.  —  The  number  of  terms  123  and  the  ratio  2. 
1st.  term  2  1st.  power. 
2 

4  2d.  power, 
2 

8  3d.  power, 
2 

16  4th.  power, 
2 

32  5th.  power 
32 


1024  10th.  power. 
2 


2048  llth.  power,  or  one  less  than  the 
2  1st.  term.  [number  of  terms. 

4096 

2  the  ratio. 

8192 

2  subtract  the  1st.  term. 

1)8190  1,  is  one  less  than  the  ratio. 

$81  90  Answer. 

£.  Bought  10  Ibs.  of  tea,  and  paid  2  cents  for  the  first 
pound,  6  for  the  second,  18  for  the  third,  &c.  What  did 
tne  whole  cost?  Ans.  $590.48. 


SEOMETRICAL    PROGRESSING  157 

3.  The  first  term  in  a  sum  is  1,  (he  whole  number  of 
terms  10,  and  the  ratio  2;  what  is  the  greatest  term, 
and  the  sum  of  all  the  terms? 

Ans.  The  greatest  term  is  512,  and  the  sum  of  the 
terms  1023. 

4.  What  debt  may  be  discharged  in  12  months,  by 
paying  1  dollar  the  first  month,  2  dollars  the  second 
month,  4  the  third  month,  and  so  on,  each  succeeding 
payment  being  double  the  last;  and  what  will  be  the 
amount  of  the  last  payment? 

Ans.  the  debt  is  $4095,  and  the  last  payment  $2048. 

5.  A  father  whose  daughter  was  married  on  a  new 
3rear'$  day,  gave  her  one  cent,  promising  to  triple  it  on 
the  first  day  of  each  month  in  the  year:  what  was  the 
amount  of  her  portion?  Ans.  $2657.20. 

6.  One  Sessa,  an  Indian,  having  invented  the  game  of 
chess,  shewed  it  to  his  prince,  who  was  so  delighted  with 
it,  that  he  promised  him  any  reward  he  should  ask; — 
upon  which  Sessa  requested  that  he  might  be  allowed 
one  grain  of  wheat  for  the  first  square  on  the  chess 
board,  2  for  the  second.  4  for  the  third,  and  so  on,  doub- 
ling continually,  to  64,  the  whole  number  of  squares. — 
JNTow,  supposing  a  pint  to  contain  7680  of  these  grains, 
and  one  quarter  or  8  bushels  to  be  worth  27s.  6d.,  it  is 
-required  to  compute  the  value  of  all  the  wheat? 

£64481488296. 

7.  What  sum  would  purchase  a  horse  with  4  shoes,  and 
eight  nails  in  each  shoe,  at  one  farthing  for  the  first  nail, 
&  halfpenny  for  the  second,  a  penny  for  the  third,  &c., 
doubling  to  the  last?  Ans. "£4473924.  5s.  3ieJ. 

8.  A  merchant  sold  15  yards  of  satin,  the  first  yard 
for  Is.  the  second  for  2s.  the  third  for  4s.  the  fourth  for 
8s.  &c.;  what  was  the  price  of  the  15  yards? 

Ans.  £1638.  7s. 

9.  Bought  30  bushels  of  wheat,  at  2d.  for  the  first 
bushel,  4d.  for  the  second,  8d.  for  the  third,  &c. ;  what 
does  the  whole  amount  to,  and  what  is  the  price  per 
bushel  on  an  average? 

\ £8947848.  10s.  6d.  Amount. 
'•  |  £298261.  12s.  4d.  per  bushel. 
14 


158  PERMUTATION. 

PERMUTATION. 

Permutation  is  used  to  show  how  many  ways  things 
may  be  varied  in  place  or  succession. 

RULE. 

Multiply  all  the  terms  of  the  series  continually,  from 
1  to  the  given  number  inclusive;  and  the  last  product 
will  be  the  answer  required. 

EXAMPLES. 

1.  How  many  changes  can  be  made  with  8  letters  of 
the  alphabet? 

1X2X3X4X6X6X7X8=40320  Answer. 

2.  In  how  many  different    positions  can   12  persons 
place  themselves  round  a  table? 

1X2X3X4X5X6X7X8X9X10X11X12  = 

479001600  Ans. 

3.  How  many  changes  may  be  made  with  the  alpha- 
bet? Ans.  620448401733239439360000, 


SKETCH  OF  MENSURATION, 

OF    PLANES   AND   SOLIDS.* 

Planes,  surfaces,  or  superficies,  are  measured  by  the 
inch,  foot,  yard,  &c.,  according  to  the  measures  used  by 
different  artists.  A  superficial  foot  is  a  plane  or  surface 
of  one  foot  in  length  and  breadth,  without  reference  to 
thickness.  Solids  are  measured  by  the  solid  inch,  foot, 
yard,  &c.;  thus,  1728  solid  inches,  that  is  12X12X12 
make  one  cubicle  or  solid  foot.  Solids  include  all  bodies 
which  have  length,  breadth  and  thickness. 

ARTICLE   I. 
To  measure  a  square  having  equal  sides. 

RULE. 

Multiply  any  one  side  of  the  square  by  itself,  and  the 
product  will  be  the  area,  or  superficial  contents,  in  feet, 
yards,  or  any  other  measure,  according  to  the  measure 
used  in  measuring  the  sides. 

*  Planes  are  the  same  as  superficies,  or  surfaces. 


SKETCH    OF    MENSURATION. 


159 


EXAMPLE. 

Let  A,  B,  C  and  D  represent  a  square,  having  equal 
sides  each  measuring  20  feet.  Multiply  the  length  of 
one  side  by  itself,  thus — 


20     feet. 
20  feet. 

Ans.  400  square  feet. 


ARTICLE    II. 

To  measure  the  plane  or  surface  of  a  parallelogram. 

RULE. 

Multiply  the  length  hy  the  breadth — the  product  will 
be  the  superficial  contents. 

EXAMPLE. 

Let  A,  B,  C  and  D  represent  a  parallelogram  whose 
length  is  40  yards,  and  breadth  15  yards. 

yards. 


breadth   15 


length     40  yards. 


Ans.  600  square  yards.  C 


Note. — The  contents  of  boards  and  other  articles 
which,  are  measured  by  feet,  &c.,  may  be  easily  found 
by  Duodecimal  fractions. 

ARTICLE    III. 

To  measure  the  plane  or  surface  of  a  triangle. 

RULE. 

Multiply  the  base  by  half  the  perpendicular,  if  it  be 
a  right  angled  triangle,  and  the  product  will  be  the  area, 
or  superficial  contents;  or  multiply  the  base  and  perpen- 
dicular together,  and  half  the  product  will  be  the  area. 
But  if  it  be  an  oblique  angled  triangle,  multiply  half 
the  length  of  the  base  by  a  perpendicular  let  fall  on  the 
base  from  the  angle  opposite  to  it,  and  the  product  will 
be  the  area. 


160 


SKETCH  OF  MENSURATION, 


EXAMPLES. 

1.  Let  C,  H  and  G  represent  a  right  angled  triangle, 
having  the  right  angle  at  G ;  the  base  C  G  being  40 
feet,  and  the  perpendicular  H  G,  28  feet. 


No.  1. 


14     feet,  or  half  the  perpendicular. 
40  feet,  or  the  base. 

560  feet — the  area. 


2.  Let  B,  C  and  D  represent  an  oblique  angled  tri- 
angle; the  length  of  the  base  B  D  being  80  feet,  and 
the  perpendicular  C  E,  28  feet. 

No.  2. 

28     the  perpendicular.  C 

40  half  the  base. 

1120  Answer. 


Mote. — Right  angled  triangles  are  such  as  have  one 
angle  like  the  corner  of  a  square,  and  which  is  called 
the  right  angle,  containing  90  degrees;  as  the  angle  G 
in  the  triangle.  No.  1. — Oblique  angled  triangles  are 
such  as  have  each  of  the  angles,  either  more  or  less 
than  90  degrees,  as  in  the  triangle,  No.  2. 

ARTICLE    IV. 

To  measure  a  circle. 

Note. — Circles  are  round  figures, bounded  every  where 
by  a  circular  line,  called  the  periphery,  and  also  the  cir- 
cumference. A  line  passing  through  the  centre  is  cal- 
led the  diameter.  Half  the  length  of  the  diameter  is 
called  the  radius. 

The  diameter  may  be  found  by  the  circumference, 
-thus — As  22  is  to  7  so  is  the  circumference  to  the  diam 
eter;  and  in  like  manner  may  the  circumference  be 


SKETCH    OF    MENSURATION.  161 

found  by  the  diameter;  for,  as  7  is  to  22,  so  is  the  di- 
ameter to  the  circumference. 

ARTICLE    V. 

To  find  the  superficial  contents,  or  area,  of  a  circle. 

RULE. 

Multiply  half  the  circumference  by  half  the  diame- 
ter, and  the  product  will  be  the  answer.  Or,  multiply 
the  square  of  the  diameter  by  .7854;  or  multiply  tlie 
square  of  the  circumference  by  .07958,  and  in  either 
case  the  product  will  be  the  answer. 

EXAMPLE. 

How  many  square  feet  are  contained  in  a  circle  whose 
circumference  is  44  feet,  and  whose  diameter  is  14  feet? 
22  half  the  circumference. 
7  half  the  diameter. 

154  square  feet.     Answer. 

The  same  may  be  done  by  multiplying  the  diameter 
find  circumference  together,  and  dividing  the  product 
by  4,  thus,  44X  14=61 6-;-4=  154.  Answer. 

ARTICLE    VI, 

To  measure  the  surface  of  a  globe  or  sphere. 

RULE. 

Multiply  the  circumference  by  the  diameter,  the  pro- 
duct will  be  the  surface,  or  area. 

EXAMPLES. 

1.  What  are  the  superficial  contents  of  a  globe  whose 
circumference  is  220  feet,  and  whose  diameter  is  70 
feet  ?  220  X  70=  1 5400  square  feet.     Ans. 

2.  How  many  square  miles  are  contained  on  the  sur- 
face of  the  whole  earth,  or  globe,  which  we  inhabit? 

The  circumference  of  the  earth  is  estimated  to  be 
25020  miles,  and  the  diameter,  7964,  nearly. 
Then,  25020X7964=199259280. 

Ans.  199259280  square  miles. 
14* 


162  SKETCH  OF  MENSURATION,- 

ARTICLE    VII. 

To  find  the  solid  contents  of  a  cube* 

RULE. 

Multiply  the  length  of  one  side  by  itself,  and  multiply 
the  product  by  the  same  length,  that  is,  by  the  same 
multiplier;  the  last  product  will  be  the  solid  contents  of 
the  cube. 

EXAMPLES. 

1.  How  many  solid  feet  are  contained  in  a  cube,  or 
solid  block  of  6  equal  sides,  each  side   being  3  feet  in 
length,  and  3  in  breadth? 

3X3X3=27  solid  or  cubick  feet.     Ans. 

When  the  contents  are  required  of  right  angled  solids, 

whose  length,  breadth,  &c.,  are  not  equal:  multiply  the 

length  by  the  breadth,  and  that  product  by  the  thick- 

nes* — the  product  will  be  the  answer 

2.  Required  the  contents  of  a  load  of  wood,  whose 
length  is  8  feet,  breadth  4  feet,  and  height  or  thickness 
4  feet.  8X4X4=128  solid  feet,  or  1  cord.     Ans. 

3.  Required  the  contents  of  a  stone  i6£  feet  in  length, 
\\  in  hrf.adth,  and  1  foot  in  thickness. 

16.5X1.5X1  =  24.75  solid  feet,  or  1  perch.     Ans, 
Note, — Solids  whose  dimensions  are  in  feet  and  inches, 
are  more  easily  measured  by  Duodecimals. 

ARTICLE    VIII. 

To  find  the  contents  of  a  prism. 

A  prism  is  an  angular  figure,  generally  ®f  three  equal 
sides,  whose  ends  are  in  the  form  of  triangles.  It  re- 
sembles a  fife  of  three  sides,  whose  whole  length  is  of 
equal  bigness. 

RULE. 

Find  the  area  or  superficial  contents  of  one  end  as  of 
any  other  triangle,  then  multiply  the  area  by  the  length 
of  the  prism,  and  the  product  will  be  the  solidity. 

EXAMPLE. 

What  are  the  solid  contents  of  a  prism,  the  sides  of 
the  triangles  of  which  measure  13  inches,  the  perpen 

*  A  cwbe  »  a  solid  body  of  equal  sides,  each  of  which  is  an  exact  square. 


SKETCH    OF    MENSURATION.  163 

dicular  extending  from  one  of  its  angles  to  its  opposite 
side,  12  inches,  and  its  length  18  inches? 
13X12=156-^-2=78X18—1404  cubick  inches.     Ans, 

ARTICLE    IX. 

To  find  the  contents  of  a  cylinder. 
A  cylinder  is  a  long  round    body,  all  its  length  being 
of  equal  bigness,  like  a  round  ruler. 

RULE. 

Find  the  area  of  one  end,  by  the  rule  for  finding  the 
area  of  a  circle,  then  multiply  it  by  the  length,  and  the 
product  will  be  the  answer. 

EXAMPLE. 

What  is  the  solidity  of  a  cypher,  the  area  of  one  end 
of  which  contains  2.40  square  feet,  and  its  length  being 
12.5  feet?  2.40  X  12.5=30  solid  feet.  Ans. 

ARTICLE    X. 

To  find  the  solid  contents  of  a  round  stick  of  timber,  which 
is  of  a  true  taper  from  the  larger  to  the  smaller  end. 

RULE. 

Find  the  area  of  both  ends;  add  the  two  areas  to^ 
gether,  and  reserve  the  sum;  multiply  the  area  of  the 
larger  end  by  the  area  of  the  smaller  end,  extract  the 
square  root  of  the  product,  add  the  root  to  the  reserved 
sum,  then  multiply  this  sum  by  one  third  the  length  of 
the  stick,  and  the  product  will  be  the  solidity. 

Note. — As  this  method  requires  considerable  labour, 
the  following  has  been  preferred  for  common  use,  though 
not  quite  so  accurate. 

RULE. 

Girt  the  stick  near  the  middle,  but  a  little  nearer  to 
the  larger  than  to  the  smaller  end;  this  will  give  the 
circumference  at  that  place.  Find  the  diameter  by  the 
circumference;  multiply  the  circumference  and  diame- 
ter together;  then  multiply  one  fourth  of  the  product 
by  the  length,  and  the  answer  will  be  nearly  the  solid 
contents. 

EXAMPLE. 

What  is  the  solidity  of  a  round  stick  of  timber  that 


164  PRACTICAL  QUESTIONS. 

is  10  feet  long,  and  its  circumference  near  the  middle  is 
2.61  feet? 

As  22    :  7    :    :  2.61    :  .83  diameter, 
cir.  diam.  length,  feet. 

2.16 X. 83=2. 1663-^4^.5415X10=^ 5  41 50. 

Ans.  5.4 160  solid  feet. 

ARTICLE    XI. 

To  find  the  solid  contents  of  a  globe. 

RULE. 

Multiply  the  cuhe  of  the  diameter  by  .5236,  the  pro- 
duct will  be  the  solid  contents.  Or,  multiply  the  super- 
ficial contents,  or  surface,  hy  one  sixth  part  of  the  sur- 
face. Or,  multiply  the  cube  of  the  diameter  by  11. 
and  divide  the  product  by  21 — in  either  case  the  pro- 
duct will  be  the  solidity. 

EXAMPLES. 

1.  What  are  the  solid  contents  of  a  globe  whose  di- 
ameter is  14  inches? 

14X  14  X  14=2744X. 5236=  1436.7584 

cubick  inches.     Ans. 

2.  How  many  solid  miles  are  contained  in  the  earth, 
or  globe,  which  we  inhabit? 

Suppose  the  diameter  to  be  7954  miles;  then,  7954X 
7954X7954=503218686664  thecubenf  the  earth'saxis, 
or  diameter;  then, 

503218686664X.5236=263485304337 

cubick  miles.     Ans. 

Note. — The  solidity  of  a  globe  may  be  found  by  the 
circumference,  thus — Multiply  the  cube  of  the  circum- 
ference by  .016887 — the  product  will  be  the  contents. 


PRACTICAL  QUESTIONS. 

1.  A  cannon  ball  goes  about  1500  feet  in  a  second  of 
time.  Moving  at  that  rate,  what  time  would  it  take  in 
going  from  the  earth  to  the  sun;  admitting  the  dis- 
tance to  be  100  millions  of  m'les,  and  the  year  to  con- 
tain 365  days,  6  hours?  Ans. 


PRACTICAL    QUESTIONS.  iGc/ 

2.  A  young-  man  spent  j  of  his  fortune  in  8  months,  f 
of  the  remainder  in  12  months  more,  after  which  he 
had  £410  left.     What  was  the  amount  of  his  fortune? 

Ans.  £966  13s.  4d. 

3.  What  number  is  that,  from  which  if  you  take  %  of 
J,  and  to  the  remainder  add  •£$  of  -^ ,  the  sum  will  be 
10?  Ans.  KVAV- 

4.  What  part  of  3,  is  a  third  part  of  2?  Ans.  f. 

5.  If  20  men  can  perform  a  piece  of  work  in  12  days, 
how  many  will  accomplish  another  thrice  as  large,  in 
one  fifth  of  the  time?  Ans.  300. 

6.  A  person  making  his  will,  gave  to  one  child  ±%  of 
his  estate,  and  the  re*t  to  another.     When  these  lega- 
cies were  paid,  the  one  proved  to  be  £600  more  than 
the  other.     What  was  the  worth  of  the  whole  estate? 

Ans.  £2000. 

7.  The  clocks  of  Italy  go  on  to  24  hours;  how  many 
strokes  do  they  strike  in  one  complete  revolution  of  the 
index?  Ans.  300. 

8.  What  quantity  of  water  must  be  added  to  a  pipe 
of  wine,  valued  at  £33.  to  bring  the  first  cost  to  4s.  6d. 
per  gallon?  Ans.  20|  gallons. 

9.  A  younger  brother  received  £6300,  which  was  J 
of  his  elder  brother's  portion.     Wrhat  was  the  whole  es- 
tate? Ans.  £14400. 

10.  WThat  number  is  that  which  being  divided  by  2,  or 
3,  4,  5,  or  6,  will  leave  1  remainder,  but  which  if  di- 
vided by  7  will  leave  no  remainder?  Ans.  72L 

11.  What  is  the  least  number  that  can  be  divided  by 
the  nine  digits  without  a  remainder?  Ans.  2520. 

12.  How  many  bushels  of  wheat,  at  $1.12  per  bushel, 
can  I  have  for  $81.76?  Ans.  73, 

13.  What  will  27  cwt.  of  iron  come  to,  at  $4.56  per 
cwt.?  Ans.  $123.12. 

14.  When  a  man's  yearly  income  is  949  dollars,  how 
much  is  it  per  day?  Ans.  $2.60, 

15.  My  factor  sends  me  word  he  has  bought  goods  to 
the  value  of  £500.  13s.  6d.  upon  my  account,*  what  will 
his  commission  come  to  at  3£  per  cent.? 

Ans.  £17.  10s.  5Jd 


166  PRACTICAL    QUESTIONS. 

16.  How  many  yards  of  cloth,  at  17s.  6d.  per  yard, 
can  I  have  for  13  cwt.  2  qrs.  of  wool,  at  14d.  per  lb.? 

Ans.  100  yards.  3}  qrs. 

17.  There  is  a  cellar  dug  that  is  12  feet  every  way, 
in  length,  breadth,  and  depth;  how  many  solid  feet  of 
earth  were  taken  out  of  it?  Ans.  1728. 

18.  If  |  of  an  ounce  cost  £  of  a  shilling,  what  will  £ 
of  a  lb.  cost?  Ans.  17s.  6d. 

19.  If  |  of  a  gallon  cost  f  of  a  £.  what  will  f  of  a  tun 
cost?  Ans.  £105. 

20.  If  |  of  a  ship  be  worth  £3740,  what  is  the  worth 
of  the  whole?  Ans.  £9973.  6s.  8d. 

21.  What  is  the  commission  on  $2176.50,  at  21  per 
cent.?  Ans.  $54.411. 

22.  In  a  certain  orchard  |  of  the  trees  bear  apples,  i 
pears,  £  plums,  60  of  them  peaches,  and  40  cherries; 
how  many  trees  are  in  the  orchard?  Ans.  1200. 

23.  If  A  travel  by  mail  at  the  rate  of  8  miles  an  hour, 
and  when  he  is  50  miles  on  his  way,  B  start  from  the 
same  place  that  A  did,  and    travel  on  horseback  the 
same  road  at  10  miles  an  hour,  how  long  and  luow  far 
will  B  travel  to  come  up  with  A? 

Ans.  25  hour?,  and  250  miles. 

24.  Bought  a  quantity  of  cloth  for  750  dollars,  &  of 
which  I  found  to  be  inferior  which  1  had  to  sell  at  1  dol- 
lar 25  cents  per  yard,  and  by  this  I  lost   100  dollars: 
what  must  I  sell  the  rest  at  per  yard  that  1  shall  lose 
nothing  by  the  whole?  Ans.  $3.15if . 

25.  If  the  Earth  goes  round  the  sun  in  365  days,  5 
hours,  48  minutes,  49  seconds,  and  its  distance  from  the 
sun  95000000  miles,  what  must  be  the  distance  of  the 
planet  Mercury  from  the  Sun,  admitting  the  time  of  its 
revolution   round  the  Sun  to  be  87  days,  23  hours,  15 
minutes,  40  seconds? 

Note. — The  planets  descrihe  equal  areas  in  equal 
time?:  therefore,  as  the  square  of  the  time  of  the  revo- 
lution of  one  planet,  round  the  Sun,  is  to  the  square  of 
the  time  of  the  revolution  of  any  other  planet,  so  is  the 
cube  of  the  distance  of  one  planet  from  the  Sun,  to  the 
cube  of  the  distance  of  any  other  from  the  Sun, 


PLATE 

10  BE  USED  IN  STUDYING    VULGAR    FRACTIONS. 


I   I 


In  using;  the  Fractional  Plate,  the  student  must  count  the  white 
spans,  and  not  the  black  lines.  The  first  row  of  squares,  or  white 
spaces,  at  the  top,  are  whole  numbers ;  the  second  row  is  divided 
into  halves ;  the  third,  into  thirds,  and  so  on  from  the  top  to  the 
bottom.  Thus  it  may  be  shown  at  one  glance,  that  7  halves 
.make  three  a;id  a  half,  or  that  8  thirds  make  2  and  2  thirds,  &e. 


CONTENTS. 


Numeration, -        -        - 

Simple  Addition,     ------- 

Simple  Subtraction, 

Simple  Multiplication,    ------  15 

Simple  Division,      -------  20 

•Federal  Money, 28 

Tables  of  Money,  Weights,  Measures,  £c.,      - 

Compound  Addition,       ------  43 

Compound  Subtraction,  ------  48 

Compound  Multiplication,       -----  52 

Compound  Division,        ------  60 

Reduction,      --------  64 

Exchange,       --------  71 

Vulgar  Fractions,    -------  76 

Decimal  Fractions,          ------  93 

Duodecimals,           -------  101 

Single  Rule  of  Three, 105 

Doable  Rule  of  Three,    -         -         -         -         -        -  111 

Practice, 113 

Fellowship,     -                           121 

Tare  and  Tret, 123 

Barter, 126 

Siuinle  Interest, 127 

Compound  Interest,         -        -        -        -         -         -  134 

Insurance,  Commission  and  Brokerage,   -         -         -  135 

Discount,         --------  136 

Equation, 137 

Loss  and  Gain, -        -  138 

Involution,      -         -         -        -        -        -        -         -  139 

Evolution,       --------  140 

.Square  Root,  --------  ib. 

Cube  Root, 143 

Biquaratc  Root, 145 

Alli'/Jition, 146 

Position, 149 

Double  Position, 151 

Arithmetical  Progression,         -  153 

Geometrical  Progression,          -----  155 

fVni;  itation, --  158 

Sketch  of  Mensuration, ib. 

Practical  Questions,         .-----  164 


02426 


iv?308O89 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


